Boundary Value Problems and Markov Processes: Functional Analysis Methods for Markov Processes [3 ed.] 3030487873, 9783030487874

Table of contents :
Preface to the Third Edition
Preface to the Second Edition
Contents
1 Introduction and Main Results
1.1 Historical Perspective of Feller's Approach to Brownian Motion
1.2 Formulation of the Problem and Statement of Main Results
1.2.1 The Differential Operator Case
(I) Analytic Semigroups in the Lp Topology
(II) Analytic Semigroups in the Topology of Uniform Convergence and Feller Semigroups
1.2.2 The Integro-Differential Operator Case
(I) Unique Solvability Theorems for Waldenfels Integro-Differential Operators
(II) Analytic Semigroups in the Lp Topology
(III) Analytic Semigroups in the Topology of Uniform Convergence and Feller Semigroups
1.3 Summary of the Contents
1.4 An Overview of Main Theorems
1.5 Notes and Comments
Part I Analytic and Feller Semigroups and Markov Processes
2 Analytic Semigroups
2.1 Analytic Semigroups via the Cauchy Integral
2.2 Generation Theorem for Analytic Semigroups
2.3 Remark on the Resolvent Estimate (2.2)
2.4 Notes and Comments
3 Markov Processes and Feller Semigroups
3.1 Continuous Functions and Measures
3.1.1 Space of Continuous Functions
3.1.2 Space of Signed Measures
3.1.3 The Riesz–Markov Representation Theorem
3.1.4 Weak Convergence of Measures
3.2 Elements of Markov Processes
3.2.1 Definition of Markov Processes
3.2.2 Markov Processes and Markov Transition Functions
3.2.3 Path Functions of Markov Processes
3.2.4 Stopping Times
3.2.5 Definition of Strong Markov Processes
3.2.6 Strong Markov Property and Uniform Stochastic Continuity
3.3 Markov Transition Functions and Feller Semigroups
3.4 Generation Theorems for Feller Semigroups
3.5 Reflecting Diffusion
3.6 Local Time on the Boundary for the Reflecting Diffusion
3.7 Notes and Comments
Part II Pseudo-Differential Operators and Elliptic Boundary Value Problems
4 Lp Theory of Pseudo-Differential Operators
4.1 Function Spaces
4.1.1 Hölder Spaces
4.1.2 Lp Spaces
4.1.3 Fourier Transforms
4.1.4 Tempered Distributions
4.1.5 Sobolev Spaces
4.1.6 Besov Spaces
4.1.7 General Sobolev and Besov Spaces
4.1.8 Sobolev's Imbedding Theorems
4.1.9 The Rellich–Kondrachov theorem
4.2 Seeley's Extension Theorem
4.3 Trace Theorems
4.3.1 Sectional Traces
4.3.2 Jump Formulas
4.4 Fourier Integral Operators
4.4.1 Symbol Classes
4.4.2 Phase Functions
4.4.3 Oscillatory Integrals
4.4.4 Definitions and Basic Properties of Fourier Integral Operators
4.5 Pseudo-Differential Operators
4.5.1 Definitions and Basic Properties of Pseudo-Differential Operators
4.5.2 Lp Boundedness of Pseudo-Differential Operators
4.5.3 Pseudo-Differential Operators on a Manifold
4.5.4 Hypoelliptic Pseudo-Differential Operators
4.5.5 Distribution Kernel of Pseudo-Differential Operators
4.6 Elliptic Pseudo-differential Operators and their Indices
4.6.1 Pseudo-Differential Operators on Sobolev Spaces
4.6.2 The Index of an Elliptic Pseudo-Differential Operator
4.7 Functional Calculus for the Laplacian via the Heat Kernel
4.8 Notes and Comments
5 Boutet de Monvel Calculus
5.1 The Spaces H, H+ and H-
5.2 Transmission Property of Pseudo-Differential Operators
5.3 Trace, Potential and Singular Green Operators on Rn+
5.3.1 Potential Operators on Rn+
5.3.2 Trace Operators on Rn+
5.3.3 Singular Green Operators on Rn+
5.3.4 Boundary Operators on Rn-1
5.4 Historical Perspective of the Wiener–Hopf Technique
5.5 Notes and Comments
6 Lp Theory of Elliptic Boundary Value Problems
6.1 Classical Potentials and Pseudo-Differential Operators
6.1.1 Single and Double Layer Potentials
6.1.2 The Green Representation Formula
6.1.3 Surface and Volume Potentials
6.2 Dirichlet Problem
6.3 Formulation of the Boundary Value Problem
6.4 Special Reduction to the Boundary
6.5 Boundary Value Problems via Boutet de Monvel Calculus
6.5.1 Boundary Space Bs-1-1/p,p(∂Ω)
6.5.2 Index Formula of Agranovič–Dynin Type
6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion
6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem
6.7.1 Unique Solvability of the Dirichlet Problem
6.7.2 A Characterization of the Resolvent set of the Dirichlet Problem
6.8 Notes and Comments
Part III Analytic Semigroups in Lp Sobolev Spaces
7 Proof of Theorem 1.2
7.1 Boundary Value Problem with Spectral Parameter
7.2 Proof of the A Priori Estimate (1.7)
7.3 Notes and Comments
8 A Priori Estimates
8.1 A Priori Estimates via Agmon's Method
8.2 Notes and Comments
9 Proof of Theorem 1.4
9.1 Proof of Theorem 1.4, Part (i)
9.1.1 Proof of Proposition 9.2
9.2 Proof of Theorem 1.4, Part (ii)
9.3 Notes and Comments
Part IV Waldenfels Operators, Boundary Operators and Maximum Principles
10 Elliptic Waldenfels Operators and Maximum Principles
10.1 Borel Kernels and Maximum Principles
10.1.1 Linear Operators having Positive Borel Kernel
10.1.2 Borel Kernels and Pseudo-Differential Operators
10.2 Maximum Principles for Elliptic Waldenfels Operators
10.2.1 Weak Maximum Principle
10.2.2 Strong Maximum Principle
10.2.3 Hopf's Boundary Point Lemma
10.3 Notes and Comments
11 Boundary Operators and Boundary Maximum Principles
11.1 Ventcel'–Lévy Boundary Operators
11.2 Positive Boundary Maximum Principles
11.2.1 Boundary Maximum Principles for Ventcel'–Lévy operators
11.2.2 Boundary Maximum Principles for Ventcel' operators
Proof of Theorem 11.4, Part (i)
Proof of Theorem 11.4, Part (ii)
11.3 Notes and Comments
Part V Feller Semigroups for Elliptic Waldenfels Operators
12 Proof of Theorem 1.5 - Part (i) -
12.1 Space C0(D M)
12.2 Proof of Part (i) of Theorem 1.5
12.3 Notes and Comments
13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6
13.1 General Existence Theorem for Feller Semigroups
13.2 Feller Semigroups with Reflecting Barrier
13.3 Proof of Theorem 1.6
13.4 Proof of Part (ii) of Theorem 1.5
13.5 Notes and Comments
14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11
14.1 Existence and Uniqueness Theorem in Hölder Spaces
14.2 Proof of Theorem 1.8
14.3 Proof of Theorem 1.9
14.4 Proof of Theorem 1.10
14.5 Proof of Theorem 1.11
14.6 Minimal Closed Extension W
14.7 Notes and Comments
15 Path Functions of Markov Processes via Semigroup Theory
15.1 Basic Definitions and Properties of Markov Processes
15.2 Path-Continuity of Markov Processes
15.3 Path-Continuity of Feller Semigroups
15.4 Examples of Multi-dimensional Diffusion Processes
15.4.1 The Neumann Case
15.4.2 The Robin Case
15.4.3 The Oblique Derivative Case
15.5 Notes and Comments
Part VI Concluding Remarks
16 The State-of-the-Art of Generation Theorems for Feller Semigroups
16.1 Formulation of the Problem
16.2 Statement of Main Results
16.2.1 The Transversal Case
16.2.2 The Non-Transversal Case
16.2.3 The Lower Order Case
16.3 Notes and Comments
References
Index
References
Index

Citation preview

Lecture Notes in Mathematics  1499

Kazuaki Taira

Boundary Value Problems and Markov Processes Functional Analysis Methods for Markov Processes Third Edition

Lecture Notes in Mathematics Volume 1499

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Advisory Board: Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Camillo De Lellis, IAS, Princeton, NJ, USA Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane M´ezard, IMJ-PRG, Paris, France Mark Podolskij, Aarhus University, Aarhus, Denmark Sylvia Serfaty, NYU Courant, New York, NY, USA Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications – quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative.

More information about this series at http://www.springer.com/series/304

Kazuaki Taira

Boundary Value Problems and Markov Processes Functional Analysis Methods for Markov Processes Third Edition

Kazuaki Taira Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki Japan

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-48787-4 ISBN 978-3-030-48788-1 (eBook) https://doi.org/10.1007/978-3-030-48788-1 Mathematics Subject Classification: 47D07, 35J25, 47D05, 60J35, 60J60 © Springer Nature Switzerland AG 1991, 2009, 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my parents, Yasunori Taira (1915–1990) and Yasue Taira (1918–2014)

Preface to the Third Edition

This book is an expanded and revised version of a set of lecture notes for the graduate courses given by the author at the University of Tsukuba (1988– 1995 and 1998–2009), at Hiroshima University (1995–1998) and at Waseda University (2009–2016), which were addressed to advanced undergraduate and beginning-graduate students with an interest in functional analysis, partial differential equations and probability. It provieds an easy-to-read reference describing a link between functional analysis, partial differential equations and probability via Markov processes. The functional analytic approach to Markov processes is distinguished by its extensive use of ideas and techniques characteristic of recent developments in the theory of pseudo-differential operators, which may be considered as a modern version of the classical potential theory. It should be emphasized that pseudo-differential operators provide a constructive tool to deal with existence and smoothness of solutions of partial differential equations. The full power of this very refined theory is yet to be exploited. Several recent developments in the theory of partial differential equations have made possible further progress in the (mesoscopic) study of elliptic boundary value problems and hence in the (microscopic) study of Markov processes, with a special emphasis on the (macroscopic) study of Feller semigroups in functional analysis. In addition to providing a comprehensive study of Markov processes, this book may also be considered as an accessible and careful introduction to two more advanced books: (A) Diffusion processes and partial differential equations, Academic Press, Boston, Massachusetts, 1988. ISBN: 0-12-682220-4 (B) Semigroups, boundary value problems and Markov processes, Second Edition, Springer Monographs in Mathematics (SMM), Springer-Verlag, 2014. ISBN: 978-3-662-43696-7

VIII

Preface to the Third Edition

Most mathematicians working in partial differential equations (the mesoscopic approach) are only vaguely familiar with the powerful ideas of stochastic analysis (the microscopic approach). However, the stochastic intuition which this book conveys may provide a profound insight into the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. In this new edition we explain in detail this neglected microscopic side of functional analysis, boundary value problems and probability, which is the first purpose of the book. Indeed, we re-work and expand in a different spirit the material of the book (B), even though there is a lot of overlap between the table of contents of this new edition and that of the SMM volume. The following three chapters have been revised: (1) Chapter 3: Markov Processes and Feller Semigroups. Section 3.6 is devoted to the local time on the boundary for the reflecting diffusion due to Paul L´evy. (2) Chapter 4: Lp Theory of Pseudo-Differential Operators. Section 4.6 is devoted to the functional calculus for the Laplacian via the heat kernel. (3) Chapter 6: Lp Theory of Elliptic Boundary Value Problems. Section 6.1 is a modern version of the classical potential theory in terms of pseudodifferential operators. Section 6.6 is a probabilistic approach to pseudodifferential operators via the reflecting Brownian motion. Very recently, we have been able to solve several long-standing open problems in the spectral analysis of elliptic boundary value problems, such as the hypoelliptic Robin problem and the subelliptic oblique derivative problem. In the proofs we made essential use of the Boutet de Monvel calculus, which is one of the most influential ideas in the modern history of analysis. The presentation of the new aspect of the Boutet de Monvel calculus is the second purpose of this third edition. Indeed, we carefully re-worked the classical functional analytic methods for Markov processes (due to Sato–Ueno and Bony–Courr`ege–Priouret) from the viewpoint of the Boutet de Monvel calculus, which will provide a powerful method for future research in semigroups, boundary value problems and Markov processes. The following six chapters are included in this new edition: (4) (5) (6) (7) (8) (9)

Chapter Chapter Chapter Chapter Chapter Chapter

5: Boutet de Monvel Calculus. 10: Elliptic Waldenfels Operators and Maximum Principles. 11: Boundary Operators and Boundary Maximum Principles. 14: Proofs of Theorems 1.8, 1.9, 1.10 and 1.11. 15: Path Functions of Markov Processes via Semigroup Theory. 16: Concluding Remarks.

In this third edition we mainly confined ourselves to simple but fundamental boundary conditions such as those in the Robin problem and the oblique derivative problem, which makes it possible to develop our basic machinery with a minimum of bother and also to present our principal ideas concretely and explicitly for advanced undergraduates and beginning-graduate

Preface to the Third Edition

IX

students. Moreover, this edition is amply illustrated; 137 figures, 15 tables and 9 flowcharts of proofs are provided with appropriate captions. Having read this book, a broad spectrum of readers will be able to easily and effectively appreciate the mathematical crossroads of functional analysis, boundary value problems and probability developed in the more advanced books (A) and (B). This is the third purpose of the new edition. Furthermore, this book provides a compendium for a large variety of facts from functional analysis, pseudo-differential operators and Markov processes – making it easy to quickly look up a theorem. Indeed, this book gives detailed coverage of important examples and applications. Bibliographical references are discussed primarily in the Notes and Comments at the end of each chapter. These notes are intended to supplement the text and place it in a better perspective. In preparing this monograph, I am indebted to Professor Yasushi Ishikawa, who has read and commented on portions of various preliminary drafts from the viewpoint of probability. In particular, Section 3.6 is essentially due to him. I would like to extend my warmest thanks to Professors Francesco Altomare and Elmar Schrohe, who have showed constant interest in my work since 1996. I would like to extend my hearty thanks to the staff of Springer-Verlag (Heidelberg), who have generously complied with all my wishes. Last but not least, I owe a great debt of gratitude to my family, who gave me moral support during the preparation of this book. Tsuchiura, April 2020

Kazuaki Taira

Preface to the Second Edition

This monograph is an expanded and revised version of a set of lecture notes for the graduate courses given by the author both at Hiroshima University (1995– 1997) and at the University of Tsukuba (1998–2000) which were addressed to the advanced undergraduates and beginning-graduate students with interest in functional analysis, partial differential equations and probability. The first edition of this monograph, which was based on the lecture notes given at the University of Tsukuba (1988–1990), was published in 1991. This edition was found useful by a number of people, but it went out of print after a few years. This second edition has been revised to streamline some of the analysis and to give better coverage of important examples and applications. The errors in the first printing are corrected thanks to kind remarks of many friends. In order to make the monograph more up-to-date, additional references have been included in the bibliography. This second edition may be considered as a short introduction to the more advanced book Semigroups, boundary value problems and Markov processes which was published in the Springer Monographs in Mathematics series in 2004. For graduate students working in functional analysis, partial differential equations and probability, it may serve as an effective introduction to these three interrelated fields of analysis. For graduate students about to major in the subject and mathematicians in the field looking for a coherent overview, it will provide a method for the analysis of elliptic boundary value problems in the framework of Lp spaces. This research was partially supported by Grant-in-Aid for General Scientific Research (No. 19540162), Ministry of Education, Culture, Sports, Science and Technology, Japan. Last but not least, I owe a great debt of gratitude to my family who gave me moral support during the preparation of this book. Tsukuba, March 2009

Kazuaki Taira

Contents

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI 1

Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Historical Perspective of Feller’s Approach to Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Formulation of the Problem and Statement of Main Results . . . 1.2.1 The Differential Operator Case . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Integro-Differential Operator Case . . . . . . . . . . . . . . . 1.3 Summary of the Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 An Overview of Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 8 15 20 30 32

Part I Analytic and Feller Semigroups and Markov Processes 2

Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Analytic Semigroups via the Cauchy Integral . . . . . . . . . . . . . . . . 2.2 Generation Theorem for Analytic Semigroups . . . . . . . . . . . . . . . 2.3 Remark on the Resolvent Estimate (2.2) . . . . . . . . . . . . . . . . . . . . 2.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 39 45 47

3

Markov Processes and Feller Semigroups . . . . . . . . . . . . . . . . . . 3.1 Continuous Functions and Measures . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Space of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Space of Signed Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Riesz–Markov Representation Theorem . . . . . . . . . . 3.1.4 Weak Convergence of Measures . . . . . . . . . . . . . . . . . . . . .

49 50 50 52 53 63

XIV

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3.2 Elements of Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Definition of Markov Processes . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Markov Processes and Markov Transition Functions . . . 3.2.3 Path Functions of Markov Processes . . . . . . . . . . . . . . . . . 3.2.4 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Definition of Strong Markov Processes . . . . . . . . . . . . . . . 3.2.6 Strong Markov Property and Uniform Stochastic Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Markov Transition Functions and Feller Semigroups . . . . . . . . . . 3.4 Generation Theorems for Feller Semigroups . . . . . . . . . . . . . . . . . 3.5 Reflecting Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Local Time on the Boundary for the Reflecting Diffusion . . . . . 3.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 64 68 73 74 76 78 79 84 95 97 98

Part II Pseudo-Differential Operators and Elliptic Boundary Value Problems 4

Lp Theory of Pseudo-Differential Operators . . . . . . . . . . . . . . . 103 4.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.1.1 H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.2 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1.4 Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1.6 Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.1.7 General Sobolev and Besov Spaces . . . . . . . . . . . . . . . . . . . 116 4.1.8 Sobolev’s Imbedding Theorems . . . . . . . . . . . . . . . . . . . . . . 117 4.1.9 The Rellich–Kondrachov theorem . . . . . . . . . . . . . . . . . . . . 118 4.2 Seeley’s Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3 Trace Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.1 Sectional Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3.2 Jump Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.4 Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.1 Symbol Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.2 Phase Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.4.3 Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.4.4 Definitions and Basic Properties of Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.5 Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.5.1 Definitions and Basic Properties of Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.5.2 Lp Boundedness of Pseudo-Differential Operators . . . . . . 150 4.5.3 Pseudo-Differential Operators on a Manifold . . . . . . . . . . 151

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4.5.4 Hypoelliptic Pseudo-Differential Operators . . . . . . . . . . . 154 4.5.5 Distribution Kernel of Pseudo-Differential Operators . . . 154 4.6 Elliptic Pseudo-differential Operators and their Indices . . . . . . . 156 4.6.1 Pseudo-Differential Operators on Sobolev Spaces . . . . . . 157 4.6.2 The Index of an Elliptic Pseudo-Differential Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.7 Functional Calculus for the Laplacian via the Heat Kernel . . . . 170 4.8 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5

Boutet de Monvel Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.1 The Spaces H, H + and H − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.2 Transmission Property of Pseudo-Differential Operators . . . . . . 185 5.3 Trace, Potential and Singular Green Operators on Rn+ . . . . . . . 191 5.3.1 Potential Operators on Rn+ . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.3.2 Trace Operators on Rn+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.3.3 Singular Green Operators on Rn+ . . . . . . . . . . . . . . . . . . . . 198 5.3.4 Boundary Operators on Rn−1 . . . . . . . . . . . . . . . . . . . . . . . 202 5.4 Historical Perspective of the Wiener–Hopf Technique . . . . . . . . . 203 5.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

6

Lp Theory of Elliptic Boundary Value Problems . . . . . . . . . . . 207 6.1 Classical Potentials and Pseudo-Differential Operators . . . . . . . 209 6.1.1 Single and Double Layer Potentials . . . . . . . . . . . . . . . . . . 209 6.1.2 The Green Representation Formula . . . . . . . . . . . . . . . . . . 212 6.1.3 Surface and Volume Potentials . . . . . . . . . . . . . . . . . . . . . . 213 6.2 Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6.3 Formulation of the Boundary Value Problem . . . . . . . . . . . . . . . . 234 6.4 Special Reduction to the Boundary . . . . . . . . . . . . . . . . . . . . . . . . 235 6.5 Boundary Value Problems via Boutet de Monvel Calculus . . . . 242 s−1−1/p,p (∂Ω) . . . . . . . . . . . . . . . . . . . . 242 6.5.1 Boundary Space B 6.5.2 Index Formula of Agranoviˇc–Dynin Type . . . . . . . . . . . . . 244 6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion . . . . . . 249 6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem . . . . . . . . 254 6.7.1 Unique Solvability of the Dirichlet Problem . . . . . . . . . . . 255 6.7.2 A Characterization of the Resolvent set of the Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.8 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Part III Analytic Semigroups in Lp Sobolev Spaces 7

Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.1 Boundary Value Problem with Spectral Parameter . . . . . . . . . . . 266 7.2 Proof of the A Priori Estimate (1.7) . . . . . . . . . . . . . . . . . . . . . . . 267 7.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

XVI

Contents

8

A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.1 A Priori Estimates via Agmon’s Method . . . . . . . . . . . . . . . . . . . 273 8.2 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

9

Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.1 Proof of Theorem 1.4, Part (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.1.1 Proof of Proposition 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 9.2 Proof of Theorem 1.4, Part (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Part IV Waldenfels Operators, Boundary Operators and Maximum Principles 10 Elliptic Waldenfels Operators and Maximum Principles . . . . 297 10.1 Borel Kernels and Maximum Principles . . . . . . . . . . . . . . . . . . . . 297 10.1.1 Linear Operators having Positive Borel Kernel . . . . . . . . 300 10.1.2 Borel Kernels and Pseudo-Differential Operators . . . . . . 304 10.2 Maximum Principles for Elliptic Waldenfels Operators . . . . . . . 307 10.2.1 Weak Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10.2.2 Strong Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . 310 10.2.3 Hopf’s Boundary Point Lemma . . . . . . . . . . . . . . . . . . . . . 316 10.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 11 Boundary Operators and Boundary Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 11.1 Ventcel’–L´evy Boundary Operators . . . . . . . . . . . . . . . . . . . . . . . . 326 11.2 Positive Boundary Maximum Principles . . . . . . . . . . . . . . . . . . . . 328 11.2.1 Boundary Maximum Principles for Ventcel’–L´evy operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 11.2.2 Boundary Maximum Principles for Ventcel’ operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Part V Feller Semigroups for Elliptic Waldenfels Operators 12 Proof of Theorem 1.5   - Part (i) - . . . . . . . . . . . . . . . . . . . . . . . . . . 345 12.1 Space C0 D \ M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 12.2 Proof of Part (i) of Theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 12.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6 . . . . . . . . . 359 13.1 General Existence Theorem for Feller Semigroups . . . . . . . . . . . . 360 13.2 Feller Semigroups with Reflecting Barrier . . . . . . . . . . . . . . . . . . . 376 13.3 Proof of Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Contents

XVII

13.4 Proof of Part (ii) of Theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 399 13.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11 . . . . . . . . . . . . . . . . . 401 14.1 Existence and Uniqueness Theorem in H¨older Spaces . . . . . . . . . 404 14.2 Proof of Theorem 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 14.3 Proof of Theorem 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 14.4 Proof of Theorem 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 14.5 Proof of Theorem 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 14.6 Minimal Closed Extension W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 14.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 15 Path Functions of Markov Processes via Semigroup Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 15.1 Basic Definitions and Properties of Markov Processes . . . . . . . . 439 15.2 Path-Continuity of Markov Processes . . . . . . . . . . . . . . . . . . . . . . 442 15.3 Path-Continuity of Feller Semigroups . . . . . . . . . . . . . . . . . . . . . . 449 15.4 Examples of Multi-dimensional Diffusion Processes . . . . . . . . . . 452 15.4.1 The Neumann Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 15.4.2 The Robin Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 15.4.3 The Oblique Derivative Case . . . . . . . . . . . . . . . . . . . . . . . . 462 15.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Part VI Concluding Remarks 16 The State-of-the-Art of Generation Theorems for Feller Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 16.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 16.2 Statement of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 16.2.1 The Transversal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 16.2.2 The Non-Transversal Case . . . . . . . . . . . . . . . . . . . . . . . . . . 477 16.2.3 The Lower Order Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 16.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

1 Introduction and Main Results

This book is an easy-to-read reference providing a link among functional analysis, partial differential equations and probability. In this introductory chapter, our problems and results are stated in such a fashion that a broad spectrum of readers could understand. Table 1.1 below gives a bird’s-eye view of Markov processes, Feller semigroups and elliptic boundary value problems and how these relate to each other.

1.1 Historical Perspective of Feller’s Approach to Brownian Motion In 1828 the Scottish botanist Robert Brown (1773–1858) observed that pollen grains suspended in water move chaotically, incessantly changing their direction of motion (see Figure 1.1 below). The physical explanation of this phenomenon is that a single grain suffers innumerable collisions with the randomly moving molecules of the surrounding water. A mathematical theory for Brownian motion was put forward by the German physicist Albert Einstein (1879–1955) in 1905 ([36]). Let p(t, x, y) be the probability density function that a one-dimensional Brownian particle starting at position x will be found at position y at time t. Einstein derived the following formula from statistical mechanical considerations:   (y − x)2 1 √ exp − . p(t, x, y) = 2Dt 2πDt Here D is a positive constant determined by the radius of the particle, the interaction of the particle with surrounding molecules, temperature and the Boltzmann constant. This gives an accurate method of measuring Avogadro’s

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 1

2

1 Introduction and Main Results

Probability (Microscopic) (approach)

Functional Analysis (Macroscopic approach)

Elliptic Boundary Value Problems (Mesoscopic approach)

Markov process X = (xt )

Feller semigroup {Tt }t≥0

Infinitesimal generator W =W

Markov transition function pt (·, dy)

Tt f =

 D

pt (·, dy) f (y)

Tt = e t W = e t W

Chapman and Kolmogorov equation

Semigroup property Tt+s = Tt · Ts

Waldenfels operator W=A+S

Absorption and reflection phenomena

Function space   C0 D \ M

Ventcel’ (Wentzell) condition L

Table 1.1. A bird’s-eye view of Markov processes, Feller semigroups and boundary value problems

number NA = 6, 023 · 1023 by observing particles. Einstein’s theory was experimentally tested by the French physicist Jean Perrin (1870–1942) between 1906 and 1909 ([90]). Brownian motion was put on a firm mathematical foundation for the first time by the American mathematician Nobert Wiener (1894–1964) in 1923 ([144], Knight [69]). Let Ω be the space of continuous functions ω : [0, ∞) −→ R

with coordinates xt (ω) = ω(t),

and let F be the smallest σ-algebra in Ω with respect to which all xt , t ≥ 0, are measurable. Namely, F is the smallest σ-algebra which contains all sets of the form x−1 t ([a, b)) = {ω ∈ Ω : a ≤ xt (ω) < b}

for t ≥ 0 and a < b.

Wiener constructed probability measures Px , x ∈ R, on F for which the following formula (1.1) holds true:  Px ω ∈ Ω : a1 ≤ xt1 (ω) < b1 , a2 ≤ xt2 (ω) < b2 , . . . , (1.1)  an ≤ xtn (ω) < bn

1.1 Historical Perspective of Feller’s Approach to Brownian Motion

3

0 -10 -20 -30 -40 -50 -60 -50 -40 -30 -20 -10

0

10

20

Fig. 1.1. Brownian motion



b1





b2

=

bn

... a1

a2

p(t1 , x, y1 )p(t2 − t1 , y1 , y2 ) . . .

an

p(tn − tn−1 , yn−1 , yn ) dy1 dy2 . . . dyn

for 0 < t1 < t2 < . . . < tn < ∞.

This formula (1.1) expresses the “starting afresh” property of Brownian motion that if a Brownian particle reaches a position, then it behaves subsequently as though that position had been its initial position. More precisely, let pt (x, E) = Px {ω ∈ Ω : x0 (ω) = x, xt (ω) ∈ E} be the transition probability that a Brownian particle starting at position x will be found in the set E at time t (see Figure 1.2 below). Then the above formula (1.1) expresses the idea that a transition from the position x to the set E in time t + s is composed of a transition from x to some position y in time t, followed by a transition from y to the set E in the remaining time s; the latter transition has probability ps (y, E) which depends only on y. Thus a Brownian particle “starts afresh”; this property is called the Markov property of Brownian motion. The measure Px is called the Wiener measure starting at x. Markov processes are an abstraction of the idea of Brownian motion. In the first works devoted to Markov processes, the most fundamental was the work of the Russian mathematician Andrey Nikolaevich Kolmogorov (1903–1987) in 1931 ([70]) where the general concept of a Markov transition function was introduced for the first time and an analytic method of describing Markov transition functions was proposed. From the viewpoint of analysis, the transition function pt (x, ·) is something more convenient than the Markov process itself. In fact, it can be shown that the transition functions of Markov processes generate solutions of certain

4

1 Introduction and Main Results

E

............................. ......... ... . ..... ........... ... ................ . . . . . . . . .......................... ... ...

.... ... ... .......... ... ... .......... ... . . ....... ... ... ....... ... . ....... ... ....... .... ....... ... ... ......... ... .................... ... ................ ... ............... ... ... ... ... ... . ............... ................ ................

t

• x

Fig. 1.2. An image of the transition probability pt (x, E)

parabolic partial differential equations such as the classical diffusion equation; and, conversely, these differential equations can be used to construct and study the transition functions and the Markov processes themselves. In the 1950s, the theory of Markov processes entered a new period of intensive development. We can associate with each transition function in a natural way a family of bounded linear operators acting on the space of continuous functions on the state space, and the Markov property implies that this family forms a semigroup. The Hille–Yosida theory of semigroups in functional analysis ([54], [147]) made possible further progress in the (microscopic) study of Markov processes. The (macroscopic) semigroup approach to Markov processes can be traced back to the pioneering work of the Croatian–American mathematician William Feller (1906–1970) in early 1950s. Feller [39], [40] characterized completely the analytic structure of one-dimensional diffusion processes; he gave an intrinsic (mesoscopic) representation of the infinitesimal generator of a one-dimensional diffusion process and determined all possible boundary conditions which describe the domain of the infinitesimal generator. The functional analytic approach to one-dimensional Brownian motion can be visualized in Table 1.2 below (cf. Knight [69, Chapter 3, Section 3.1]): The probabilistic meaning of Feller’s work was clarified by Eugene Borisovich Dynkin (1924–2014) [33] and [34], Kiyosi Itˆ o (1915–2008) and Henry P. McKean, Jr. [65], Daniel Ray [92] and others. One-dimensional diffusion processes are completely studied both from analytic and probabilistic viewpoints (see Ikeda–Watanabe [62], Revuz–Yor [94]). The French mathematician Paul L´evy (1886–1971) found another construction of Brownian motion, and gave a profound description of (microscopic) qualitative properties of the individual Brownian path in his book ([76]): Processus stochastiques et mouvement brownien (1948).

1.2 Formulation of the Problem and Statement of Main Results Transition function √ 2 1/ 4πt e−(x−y) /4t dy ⇑ Laplace transform ⇓

Feller semigroup 2 2 Tt = et ∂ /∂x

Dynkin ⇐==⇒ Tt f (x) = 2 √1 e−(x−y) /4t f (y)dy 4πt R

Green kernel √ √ 1/ 4α e− α |x−y|

5

⇑ ⇓

Hille–Yosida

Green operator −1 α − ∂ 2 /∂x2

Riesz–Markov

Table 1.2. Feller’s approach to one-dimensional Brownian motion

1.2 Formulation of the Problem and Statement of Main Results Now let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary (see Figure 1.3 below).

∂D n D

Fig. 1.3. The bounded domain D and the inward normal n to the boundary ∂D

In this section, we consider a second-order, elliptic Waldenfels integrodifferential operator W with real coefficients such that W u(x) = Au(x) + Su(x) (1.2) ⎛ ⎞ N N   ∂2u ∂u := ⎝ aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ ∂xi ∂xj ∂xi i,j=1 i=1 ⎛ ⎞  N  ∂u ⎝u(x + z) − u(x) − + zj (x)⎠ s(x, z) m(dz). ∂xj RN \{0} j=1 Here: (1) aij ∈ C ∞ (D), aij (x) = aji (x) for all 1 ≤ i, j ≤ N , and there exists a constant a0 > 0 such that

6

1 Introduction and Main Results N 

aij (x)ξi ξj ≥ a0 |ξ|2

for all x ∈ D and ξ ∈ RN .

i,j=1

(2) bi ∈ C ∞ (D) for all 1 ≤ i ≤ N . (3) c ∈ C ∞ (D), and c(x) ≤ 0 in D, but c(x) ≡ 0 in D. (4) s(x, z) ∈ L∞ (RN × RN ) and 0 ≤ s(x, z) ≤ 1 almost everywhere in RN × RN , and there exist constants C0 > 0 and 0 < θ0 < 1 such that • |s(x, z) − s(y, z)| ≤ C0 |x − y|

θ0

(1.3a)

for all x, y ∈ D and almost all z ∈ R , N

• s(x, z) = 0

if x ∈ D and x + z ∈ D.

(1.3b)

Probabilistically, the support condition (1.3b) implies that all jumps from D are within D. Analytically, the support condition (1.3b) guarantees that the integral operator S may be considered as an operator acting on functions u defined on the closure D (see [48, Chapter II, Remark 1.19]). (5) The measure m(dz) is a Radon measure on RN \ {0} which has a density with respect to the Lebesgue measure dz on RN , and satisfies the moment condition   2 |z| m(dz) + |z| m(dz) < ∞. (1.4) {01}

(6) Finally, we assume that (W 1)(x) = (A1)(x) + (S1)(x) = c(x) ≤ 0 and c(x) ≡ 0 in D.

(1.5)

The operator W = A + S is called a second-order, Waldenfels integrodifferential operator or simply Waldenfels operator (cf. [15], [140]). The integro-differential operator S is called a second-order, L´evy integro-differential operator which is supposed to correspond to the jump phenomenon in the closure D; a Markovian particle moves by jumps to a random point, chosen with kernel s(x, z) and Radon measure m(dz), in D (see [111], [11], [63]). The differential operator A describes analytically a strong Markov process with continuous paths in the interior D such as Brownian motion. The operator A is called a diffusion operator, and the functions aij (x), bi (x) and c(x) are called the diffusion coefficients, the drift coefficients and the termination coefficient, respectively. The L´evy integro-differential operator S is supposed to correspond to the jump phenomenon in the closure D. Namely, a Markovian particle moves by jumps to a random point, chosen with kernel s(x, z), in the interior D. The function s(x, z) is called the jump density. Therefore, the Waldenfels integro-differential operator W = A + S is supposed to correspond to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D (see Figure 1.4 below). In this context, the support condition (1.3b) implies that any Markovian particle does not move by jumps from the interior D into the outside of the

1.2 Formulation of the Problem and Statement of Main Results

7

D

Fig. 1.4. A Markovian particle moves both by jumps and continuously in the state space

closure D. On the other hand, the moment condition (1.4) imposes various conditions on the structure of jumps for the L´evy operator S. More precisely, the condition  2 |z| m(dz) < ∞ {01}

implies that the measure m(·) admits a singularity of order 1 at infinity, and this singularity at infinity is produced by the accumulation of large jumps of Markovian particles. Example 1.1. A typical example of the Radon measure m(dz) which satisfies the moment condition (1.4) is given by the formula ⎧ 1 ⎪ ⎨ N +2−ε dz for 0 < |z| ≤ 1, |z| m(dz) = 1 ⎪ ⎩ dz for |z| > 1, |z|N +ε where ε > 0. Let L be a first-order, Ventcel’ boundary condition such that Lu(x ) = μ(x )

∂u  (x ) + γ(x )u(x ) for x ∈ ∂D. ∂n

Here: (1) μ ∈ C ∞ (∂D) and μ(x ) ≥ 0 on ∂D. (2) γ ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D.

(1.6)

8

1 Introduction and Main Results

(3) n = (n1 , n2 , . . . , nN ) is the unit inward normal to the boundary ∂D (see Figure 1.3). Remark 1.1. Just as in Taira [123], we can study the boundary condition L0 of the form ∂u L0 u(x ) = μ(x ) (x ) + γ(x )u(x ), (1.6 ) ∂ν where ∂/∂ν is the conormal derivative associated with the elliptic differential operator A N  ∂ ∂ = ajk (x )nk . ∂ν ∂xj j,k=1

We remark that if μ(x ) = 0 and γ(x ) ≡ 0 on ∂D (resp. μ(x ) ≡ 0 and γ(x ) = 0 on ∂D), then the boundary condition L is essentially the so-called Neumann (resp. Dirichlet) condition. The terms μ(x )∂u/∂n and γ(x )u of the boundary condition L are supposed to correspond to reflection and absorption phenomena at the boundary ∂D, respectively. The situation may be represented schematically as in Figure 1.5 below.

D

D

∂D

∂D

absorption

reflection

Fig. 1.5. The absorption phenomenon and the reflection phenomenon

1.2.1 The Differential Operator Case First, we consider the case where S ≡ 0 in D, that is, the differential operator case:  (A − λ) u = f in D, (∗)λ Lu = ϕ on ∂D. Here Au(x) =

N  i,j=1

aij (x)

N  ∂2u ∂u (x) + bi (x) (x) + c(x)u(x) ∂xi ∂xj ∂x i i=1

1.2 Formulation of the Problem and Statement of Main Results

9

and λ is a complex parameter. We study the homogeneous problem (∗)λ in the framework of Lp Sobolev spaces. If 1 ≤ p < ∞, we let Lp (D) = the space of (equivalence classes of) Lebesgue measurable functions u(x) on D such that |u(x)|p is integrable on D. The space Lp (D) is a Banach space with the norm 

u p =

1/p |u(x)| dx . p

D

If m is a non-negative integer, we define the usual Sobolev space W m,p (D) = the space of (equivalence classes of) functions u ∈ Lp (D) whose derivatives Dα u(x), |α| ≤ m, in the sense of distributions are in Lp (D). The space W m,p (D) is a Banach space with the norm ⎛

u m,p = ⎝

 

|α|≤m

⎞1/p |Dα u(x)|p dx⎠

.

D

We remark that W 0,p (D) = Lp (D);

· 0,p = · p .

Furthermore, we let B m−1/p,p (∂D) = the space of the boundary values ϕ(x ) of functions u ∈ W m,p (D). In the space B m−1/p,p (∂D), we introduce a norm |ϕ|m−1/p,p = inf u m,p , where the infimum is taken over all functions u ∈ W m,p (D) that equal ϕ on the boundary ∂D. The space B m−1/p,p (∂D) is a Banach space with respect to this norm | · |m−1/p,p ; more precisely, it is a Besov space (cf. [2], [13], [112], [135]). Hence we have, by the trace theorem,   ∂u + γ(x )u ∈ B 1−1/p,p (∂D) for u ∈ W 2,p (D). Lu = μ(x ) ∂n ∂D It should be emphasized that problem (∗)λ is a degenerate, elliptic boundary value problem in the sense of Lopatinskii–Shapiro (see [26, Chapitre V,

10

1 Introduction and Main Results

condition (4.5)]; [61, Chapter XX, Definition 20.1.1]; [93, Chapter 3, p. 194, Definition 1]; [146, Chapter II, Condition 11.1]). This is due to the fact that the so-called Lopatinskii–Shapiro complementary condition is violated at each point x of the set M = {x ∈ ∂D : μ(x ) = 0} . (see [123, Example 6.1]). More precisely, it is easy to see that the boundary value problem (∗)λ is non-degenerate (or coercive) if and only if either μ(x ) = 0 on ∂D (the regular Robin case) or μ(x ) ≡ 0 and γ(x ) = 0 on ∂D (the Dirichlet case). The generation theorem of analytic semigroups is well established in the non-degenerate case both in the Lp topology and in the topology of uniform convergence (cf. Friedman [43], Tanabe [132], Masuda [78], Stewart [110]). In this book, under the condition that μ(x ) ≥ 0 on ∂D we shall consider the problem of existence and uniqueness of solutions of the boundary value problem (∗)λ in the framework of Sobolev spaces of Lp type, and generalize the generation theorem for analytic semigroups to the degenerate case. Our fundamental conditions on L are formulated as follows: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D. A probabilistic meaning of condition (B) is (see Figure 1.6 below) that absorption phenomenon occurs at each point of the boundary portion M = {x ∈ ∂D : μ(x ) = 0} , while reflection phenomenon occurs at each point of the boundary portion ∂D \ M = {x ∈ ∂D : μ(x ) > 0} .

D ∂D \ M = {μ > 0} M = {μ = 0} Fig. 1.6. A probabilistic meaning of condition (B)

Amann [8] studied the non-degenerate case; more precisely, he assumes that the boundary ∂D is the disjoint union of the two closed subsets M and ∂D \ M , each of which is an (N − 1) dimensional compact smooth manifold.

1.2 Formulation of the Problem and Statement of Main Results

11

We give a simple example of the functions μ(x ) and γ(x ) in a relatively compact domain D with smooth boundary ∂D in the plane R2 (N = 2):   Example 1.2. LetD = (x1 , x2 ) ∈ R2 : x21 + x22 < 1 be the unit disk with the boundary ∂D = (x1 , x2 ) ∈ R2 : x21 + x22 = 1 . For a local coordinate system x1 = cos θ, x2 = sin θ with θ ∈ [0, 2π] on the unit circle ∂D, we define functions μ(x1 , x2 ) and γ(x1 , x2 ) as follows: μ (x1 , x2 ) = μ (cos θ, sin θ) ⎧   2 1 ⎪ ⎪e π2 − 1θ 1 − e π + θ− π2 ⎪ ⎪ ⎪ ⎪ ⎨ = 12   1 2 1 ⎪ 3π π+ ⎪ ⎪ e θ− 2 1 − e π − θ−π ⎪ ⎪ ⎪ ⎩ 0

  for θ ∈ 0, π2 ,   for θ ∈ π2 , π ,   for θ ∈ π, 3π 2 ,   for θ ∈ 3π 2 , 2π ,

and γ (x1 , x2 ) = μ (x1 , x2 ) − 1 on ∂D. Here

   3π 2 , 2π . M = (cos θ, sin θ) ∈ R : θ ∈ 2

Therefore, the crucial point in our approach is how to generalize the classical variational approach to the degenerate case. We begin with the following a priori estimate (1.7) in the framework of Lp Sobolev spaces (see [121, Theorem 1.1]): Theorem 1.2. Let 1 < p < ∞. Assume that conditions (A) and (B) are satisfied. Then, for any solution u ∈ W 2,p (D) of problem (∗)λ with f ∈ Lp (D) and ϕ ∈ B 2−1/p,p (∂D) we have the a priori estimate  

u 2,p ≤ C(λ) f p + |ϕ|2−1/p,p + u p , (1.7) with a positive constant C(λ) depending on λ. Remark 1.3. Some remarks are in order. (1) It is worthwhile pointing out that the a priori estimate (1.7) is the same one for the Dirichlet condition: μ(x ) ≡ 0 and γ(x ) = 0 on ∂D (cf. [4], [77]). This rather surprising result (elliptic estimates for a degenerate problem) works, since the degeneracy occurs only for the boundary data ϕ. (2) More precisely, we can obtain an existence and uniqueness theorem for the non-homogeneous boundary value problem  Au = f in D, (∗) Lu = ϕ on ∂D in the framework of Lp Sobolev spaces, if we take S := 0 in Theorem 1.8 below.

12

1 Introduction and Main Results

(I) Analytic Semigroups in the Lp Topology First, we formulate a generation theorem for analytic semigroups in the Lp topology. We associate with problem (∗)λ an unbounded linear operator Ap : Lp (D) −→ Lp (D) in the Banach space Lp (D) into itself as follows: (a) The domain of definition D (Ap ) is the set   2,p  ∂u  + γ(x )u = 0 on ∂D . D(Ap ) = u ∈ W (D) : Lu = μ(x ) ∂n

(1.8)

(b) Ap u = Au for every u ∈ D (Ap ). Then we can prove that the operator Ap generates an analytic semigroup in the Banach space Lp (D) (see [121, Theorem 1.2]): Theorem 1.4. Let 1 < p < ∞. Assume that conditions (A) and (B) are satisfied. Then we have the following two assertions: (i) For every positive number ε, there exists a positive constant rp (ε) such that the resolvent set of Ap contains the set   Σp (ε) = λ = r2 ei θ : r ≥ rp (ε), −π + ε ≤ θ ≤ π − ε , and that the resolvent (Ap − λI)−1 satisfies the estimate   (Ap − λI)−1  ≤ cp (ε) |λ|

for all λ ∈ Σp (ε),

(1.9)

where cp (ε) is a positive constant depending on ε. (ii) The operator Ap generates a semigroup ez Ap on Lp (D) that is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2 (see Figure 1.7 below).

(II) Analytic Semigroups in the Topology of Uniform Convergence and Feller Semigroups Secondly, we formulate a generation theorem for analytic semigroups in the topology of uniform convergence. Let C(D) be the space of real-valued, continuous functions f (x) on D. We equip the space C(D) with the topology of uniform convergence on the whole D; hence it is a Banach space with the maximum norm

1.2 Formulation of the Problem and Statement of Main Results

Σp (ε)

ε

ε

|λ| = rp (ε)2

ε

13

0

Δε 0 ε

Fig. 1.7. The set Σp (ε) and the sector Δε

f ∞ = max |f (x)|. x∈D

We introduce a subspace of C(D) that is associated with the boundary condition L. We remark that the boundary condition    ∂u   + γ(x )u = 0 on ∂D Lu = μ(x ) ∂n ∂D includes the condition u=0

on M = {x ∈ ∂D : μ(x ) = 0},

if γ(x ) = 0 on M . With this fact in mind, we let     C0 D \ M = u ∈ C(D) : u = 0 on M .   The space C0 D \ M is a closed subspace of C(D); hence it is a Banach space. Furthermore, we introduce an unbounded linear operator     A : C0 D \ M −→ C0 D \ M as follows: (a) The domain of definition D (A) is the set       D (A) = u ∈ C0 D \ M : Au ∈ C0 D \ M , Lu = 0 on ∂D . (1.10) (b) Au = Au for every u ∈ D (A). Here Au and Lu are taken in the sense of distributions (see Chapter 12). Then we can prove that theoperator A generates an analytic semigroup  in the Banach space C0 D \ M ([121, Theorem 1.3]):

14

1 Introduction and Main Results

Theorem 1.5. Assume that conditions (A) and (B) are satisfied. Then we have the following two assertions: (i) For every positive number ε, there exists a positive constant r(ε) such that the resolvent set of A contains the set   Σ(ε) = λ = r2 ei θ : r ≥ r(ε), −π + ε ≤ θ ≤ π − ε , and that the resolvent (A − λI)−1 satisfies the estimate

(A − λI)−1 ≤

c(ε) |λ|

for all λ ∈ Σ(ε),

(1.11)

where c(ε) is a positive constant depending on ε.   (ii) The operator A generates a semigroup ez A on C0 D \ M that is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2 (see Figure 1.8 below).   Moreover, the operators et A t≥0 are non-negative and contractive on   C0 D \ M :   f ∈ C0 D \ M , 0 ≤ f (x) ≤ 1 on D \ M (1.12) =⇒

0 ≤ et A f (x) ≤ 1 on D \ M .

Σ(ε) |λ| = r(ε)2

ε ε

ε

0

Δε 0 ε

Fig. 1.8. The set Σ(ε) and the sector Δε

The main purpose of this book is devoted to the functional analytic approach to the problem of existence of Markov processes in probability   theory. A strongly continuous semigroup {Tt }t≥0 on the space C0 D \ M is called a Feller semigroup on thestate space  D\M if it is non-negative and contractive on the Banach space C0 D \ M . Therefore, we can reformulate assertion (1.12) of Theorem 1.5 as follows ([121, Theorem 1.4]):

1.2 Formulation of the Problem and Statement of Main Results

15

Theorem 1.6. If conditions (A) and  (B) are satisfied, then the operator A generates a Feller semigroup et A t≥0 on the state space D \ M . Theorem 1.6 generalizes Bony–Courr`ege–Priouret [15, Th´eor`eme XIX] to the case where μ(x ) ≥ 0 on the boundary ∂D (cf. [114, Theorem 10.1.3]). It is known (cf. [34], [122, Chapter 9]) that if {Tt }t≥0 is a Feller semigroup on the state space D\M , then there exists a unique Markov transition function pt (x, ·) on the space D \ M such that    Tt f (x) = pt (x, dy)f (y) for all f ∈ C0 D \ M . D\M

Furthermore, it can be shown that the function pt (x, ·) is the transition function of some strong Markov process X ; hence the value pt (x, E) expresses the transition probability that a Markovian particle starting at position x will be found in the set E at time t. The differential operator A describes analytically a strong Markov process with continuous paths in the interior D such as Brownian motion. Rephrased, Theorem 1.6 asserts that there exists a Feller semigroup on the state space D corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space D \ M until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (see Figure1.6). Remark 1.7. It is worthwhile pointing out here that the condition μ(x ) ≥ 0 and γ(x ) ≤ 0

on ∂D

is necessary in order that the operator A is the infinitesimal generator of a Feller semigroup {Tt }t≥0 on the state space D \ M (cf. [114, Section 9.5]). Our functional analytic approach to strong Markov processes may be visualized as in Table 1.3 below (see also Figure 1.9 below). 1.2.2 The Integro-Differential Operator Case Now we consider the general case, that is, the integro-differential operator case:  (W − λ) u = f in D, (∗∗)λ Lu = ϕ on ∂D. Here W u(x) = Au(x) + Su(x) ⎛ ⎞ N N 2   ∂ u ∂u := ⎝ aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ ∂x ∂x ∂x i j i i,j=1 i=1

(1.2)

16

1 Introduction and Main Results Transition function pt (x, dy)

Feller semigroup T t = et

Dynkin ⇐==⇒

⇑ Laplace transform ⇓

Tt f (x) =

Green kernel Gα (x, y)

D

pt (x, dy)f (y)

⇑ ⇓

Hille–Yosida

Green operator −1 (αI − )

Riesz–Markov

Table 1.3. A semigroup approach to strong Markov processes

⎛

⎞ ∂u ⎝u(x + z) − u(x) − + zj (x)⎠ s(x, z) m(dz) ∂x j RN \{0} j=1 

N 

and λ is a complex parameter. (I) Unique Solvability Theorems for Waldenfels Integro-Differential Operators The first purpose of this book is to prove an existence and uniqueness theorem for the following non-homogeneous boundary value problem  W u = f in D, (∗∗) Lu = ϕ on ∂D in the framework of H¨ older spaces. Due to the non-local character of the Waldenfels integro-differential operator W = A + S, we find more difficulties in the bounded domain D than in the whole space RN . In fact, when considering the Dirichlet problem in D, it is natural to use the zero extension of functions in the interior D outside of the closure D = D ∪ ∂D. This extension has a probabilistic interpretation. Namely, this corresponds to stopping the diffusion process with jumps in the whole space RN at the first exit time of the closure D. However, the zero extension produces a singularity of solutions at the boundary ∂D. In order to remove this singularity, we introduce various conditions on the structure of jumps for the Waldenfels integro-differential operator W = A + S such as conditions (1.3a), (1.3b) and (1.4). On the other hand, we introduce a subspace of C 1+θ (∂D) which is associated with the degenerate boundary condition

1.2 Formulation of the Problem and Statement of Main Results

   ∂u   + γ(x )u Lu = μ(x ) ∂n ∂D

17

on ∂D.

If conditions (A) and (B) are satisfied, we let C1+θ (∂D) := μ(x ) C 1+θ (∂D) + |γ(x )| C 2+θ (∂D)   = ϕ = μ(x )ϕ1 − γ(x )ϕ2 : ϕ1 ∈ C 1+θ (∂D), ϕ2 ∈ C 2+θ (∂D) , and define a norm |ϕ|C1+θ (∂D)   = inf |ϕ1 |C 1+θ (∂D) + |ϕ2 |C 2+θ (∂D) : ϕ = μ(x )ϕ1 − γ(x )ϕ2 . Then it is easy to verify (see the proof of [123, Lemma 6.8]) that the space C1+θ (∂D) is a Banach space with respect to the norm | · |∗1+θ . We remark that the space C1+θ (∂D) is an “interpolation space” between the spaces C 1+θ (∂D) and C 2+θ (∂D). More precisely, we have the assertions C1+θ (∂D)  C 2+θ (∂D) if μ(x ) ≡ 0 on ∂D (the Dirichlet case), = C1+θ (∂D) = C 1+θ (∂D) if μ(x ) > 0 on ∂D (the Robin case), and, for general μ(x ), the continuous injections C 1+θ (∂D) ⊂ C1+θ (∂D) ⊂ C 2+θ (∂D). The next theorem is a generalization of [117, Theorem 1] and [123, Theorem 1.1] to the integro-differential operator case : Theorem 1.8. Assume that the following three conditions (A), (B) and (H) are satisfied: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D. (H) The integral operator S satisfies conditions (1.3a), (1.3b), (1.4) and (1.5). Then the mapping (W, L) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is an algebraic and topological isomorphism for all 0 < θ ≤ θ0 . In particular, for any f ∈ C θ (D) and any ϕ ∈ C1+θ (∂D), there exists a unique solution u ∈ C 2+θ (D) of problem (∗∗).

18

1 Introduction and Main Results

(II) Analytic Semigroups in the Lp Topology The second purpose of this book is to study problem (∗∗)λ from the point of view of analytic semigroup theory in functional analysis. First, we formulate a generation theorem for analytic semigroups in the Lp topology. To do this, we associate with problem (∗∗)λ an unbounded linear operator Wp : Lp (D) −→ Lp (D) in the Banach space Lp (D) into itself as follows: (a) The domain of definition D (Wp ) is the set   D (Wp ) = u ∈ W 2,p (D) : Lu = 0 on ∂D .

(1.13)

(b) Wp u = W u for every u ∈ D (Wp ). Here W u and Lu are taken in the sense of distributions (see Chapters 12 and 13). The next theorem is a generalization of Theorem 1.5 to the integro-differential operator case: Theorem 1.9. Let 1 < p < ∞. Assume that conditions (A), (B) and (H) are satisfied. Then we have the following two assertions: (i) For every ε > 0, there exists a constant rp (ε) > 0 such that the resolvent set of Wp contains the set   Σp (ε) = λ = r2 ei ϑ : r ≥ rp (ε), −π + ε ≤ ϑ ≤ π − ε , and that the resolvent (Wp − λI)−1 satisfies the estimate   (Wp − λI)−1  ≤ cp (ε) |λ|

for all λ ∈ Σp (ε),

(1.14)

where cp (ε) > 0 is a constant depending on ε. (ii) The operator Wp generates a semigroup ez Wp on Lp (D) which is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2. (III) Analytic Semigroups in the Topology of Uniform Convergence and Feller Semigroups Secondly, we state a generation theorem for analytic semigroups in the topology of uniform convergence. We introduce a linear operator     W : C0 D \ M −→ C0 D \ M   in the Banach space C0 D \ M as follows:

1.2 Formulation of the Problem and Statement of Main Results

19

(a) The domain of definition D (W) is the set D (W) (1.15)       2,p = u ∈ C0 D \ M ∩ W (D) : W u ∈ C0 D \ M , Lu = 0 on ∂D . (b) Wu = W u for every u ∈ D (W). Here we remark that the domain D (W) is independent of N < p < ∞ (see the proof of Lemma 14.12). Then we can prove that  the operator W generates an analytic semigroup in the Banach space C0 D \ M . In other words, Theorem 1.9 remains valid   with Lp (D) and Wp replaced by C0 D \ M and W, respectively: Theorem 1.10. Let N < p < ∞. Assume that conditions (A), (B) and (H) are satisfied. Then we have the following two assertions: (i) For every ε > 0, there exists a constant r(ε) > 0 such that the resolvent set of W contains the set   Σ(ε) = λ = r2 ei ϑ : r ≥ r(ε), −π + ε ≤ ϑ ≤ π − ε , and that the resolvent (W − λI)−1 satisfies the estimate   (W − λI)−1  ≤ c(ε) |λ|

for all λ ∈ Σ(ε),

(1.16)

where c(ε) > 0 is a constant depending on ε.   (ii) The operator W generates a semigroup ez W on C0 D \ M that is analytic in the sector Δε = {z = t + is : z = 0, |arg z| < π/2 − ε} for any 0 < ε < π/2. Theorems 1.9 and 1.10 express a regularizing effect for the parabolic integro-differential operator ∂/∂t − W with the homogeneous boundary condition L (cf. [48, Chapter VIII, Theorem 3.1]). As an application of Theorem 1.8, we consider the problem of existence of Markov processes in probability theory. To do this, we define a linear operator     W : C0 D \ M −→ C0 D \ M in the Banach space as follows: (a) The domain of definition D (W) is the set D (W) (1.17)       2 = u ∈ C (D) ∩ C0 D \ M : W u ∈ C0 D \ M , Lu = 0 on ∂D .

20

1 Introduction and Main Results

(b) Wu = W u for every u ∈ D (W). The next theorem asserts that there exists a Feller semigroup on the state space D \ M corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D \ M until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (cf. [72, Theorem 5.2], [111, Theorem 2.2], [48, Chapter VIII, Theorem 3.3]): Theorem 1.11. If conditions (A), (B)and (H) are satisfied, then the operator  W is closable in the space C0 D \ M , and its minimal closed extension W is the infinitesimal generator of some Feller semigroup Tt = et W on the state space D \ M . Remark 1.12. By combining Theorems 1.11 and 1.10, we can prove that the operator W coincides with the minimal closed extension W (see Section 14.6): W = W.

(14.45)

Our functional analytic approach to strong Markov processes may be visualized as in Figure 1.9 below.

{Tt } : Feller semigroup on C0 (D \ M )

pt (x, ·) : uniform stochastic continuity + C0 -property

right-continuous Markov process

X : strong Markov process

Fig. 1.9. A functional analytic approach to strong Markov processes

1.3 Summary of the Contents This introductory chapter 1 is intended as a brief introduction to our problem and results in such a fashion that a broad spectrum of readers could understand. The contents of the book are divided into five principal parts. Chapter 2 is devoted to a review of standard topics from the theory of analytic semigroups which forms a functional analytic background for the proof of Theorems 1.4 and 1.5.

1.3 Summary of the Contents

21

Chapter 3 is devoted to a functional analytic approach to Markov processes in probability which forms a functional analytic background for the proof of Theorems 1.6 and 1.11. Section 3.1 is devoted to the Riesz–Markov representation theorem which describes an intimate relationship between measures and linear functionals (Theorems 3.7 and 3.10). Section 3.2 is devoted to a brief description of basic definitions and results about Markov processes. We give concrete important examples of Markov transition functions on the line R = (−∞, ∞), and some examples of diffusion processes on the half line R+ = [0, ∞) in which we must take account of the effect of the boundary point 0. Section 3.3 provides a brief description of basic definitions and results about Markov processes and a class of semigroups (Feller semigroups) associated with Markov processes. The semigroup approach to Markov processes can be traced back to the pioneering work of Feller [39] and [40] in early 1950s (cf. [15], [98], [116]). In Section 3.4 we prove various generation theorems of Feller semigroups by using the Hille–Yosida theory of semigroups (Theorems 3.34 and 3.36). We give two examples where it is difficult to begin with a Markov transition function and the infinitesimal generator is the basic tool of describing the process (Examples 3.16 and 3.17). Section 3.5 is devoted to the reflecting diffusion that is associated with the homogeneous Neumann problem for the Laplacian. In Section 3.6, following Sato–Tanaka [97] and Sato–Ueno [98] we introduce the notion of local time on the boundary for the reflecting diffusion constructed in Section 3.5. In Chapter 4 we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators which may be considered as a modern version of the classical potential theory. In Section 4.1 we define H¨older spaces C k+θ (Ω) and various Sobolev spaces s,p W (Ω) and H s,p (Ω) and also Besov spaces B s,p (∂Ω) on the boundary ∂Ω of a smooth domain Ω of Euclidean space Rn . It is the imbedding characteristics of Lp Sobolev spaces that render these spaces so useful in the study of partial differential equations. In the proof of Theorem 1.5 we shall make use of some imbedding properties of Lp Sobolev spaces (Theorem 4.8). Moreover, we shall need the Rellich–Kondrachov compactness theorem for function spaces of Lp type (Theorem 4.10) in the proof of Theorem 1.4. The Rellich–Kondrachov theorem is a Sobolev space version of the Bolzano–Weierstrass theorem and the Ascoli–Arzel`a theorem in calculus. In Section 4.2 we formulate Seeley’s extension theorem (Theorem 4.11), due to Seeley [103], which asserts that the functions in C ∞ (Ω) are the restrictions to Ω of functions in C ∞ (Rn ). It should be emphasized that Besov spaces B s,p (∂Ω) enter naturally in connection with boundary value problems in the framework of Sobolev spaces of Lp type. Indeed, we need to make sense of the restriction u|∂Ω to the

22

1 Introduction and Main Results

boundary ∂Ω as an element of a Besov space on ∂Ω when u belongs to a Sobolev space on the domain Ω. In Section 4.3, we formulate an important trace theorem (Theorem 4.18) that will be used in the study of boundary value problems. In Section 4.4, we present a brief description of basic concepts and results of the theory of Fourier integral operators and pseudo-differential operators. Pseudo-differential operators provide a constructive tool to deal with existence and smoothness of solutions of partial differential equations. The theory of pseudo-differential operators continues to be one of the most influential works in modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited. In Section 4.5 we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators. We formulate the Besov space boundedness theorem due to Bourdaud [16] (Theorem 4.47) in Subsection 4.5.2 and we give a useful criterion for hypoellipticity due to H¨ormander [59] (Theorem 4.49) in Subsection 4.5.4, which plays an essential role in the proof of our main results. In Subsection 4.5.5, following Coifman–Meyer [30, Chapitre IV, Proposition 1]) we state that the distribution kernel  1 ei(x−y)·ξ a(x, ξ) dξ, s(x, y) = (2π)n Rn n of a pseudo-differential operator S ∈ Lm 1,0 (R ) with symbol a(x, ξ) satisfies the estimate (Theorem 4.51)

|s(x, y)| ≤

C |x − y|m+n

for all x, y ∈ Rn and x = y.

In Section 4.6, by using the Riesz–Schauder theory we prove some of the most important results about elliptic pseudo-differential operators on a manifold and their indices in the framework of Sobolev spaces (Theorems 4.53 through 4.67). These results play an important role in the study of elliptic boundary value problems in Chapter 5. The heat kernel Kt (x) =

|x|2 1 e− 4t n/2 (4πt)

for t > 0

has many important and interesting applications in partial differential equations. In Section 4.7, by calculating various convolution kernels for the Laplacian via the Laplace transform we derive Newtonian, Riesz and Bessel potentials and also the Poisson kernel for the Dirichlet boundary value problem (Theorems 4.69, 4.70 and 4.73). In Chapter 5 we introduce the notion of transmission property due to Boutet de Monvel [19], which is a condition about symbols in the normal direction at the boundary. Elliptic boundary value problems cannot be treated directly by pseudo-differential operator methods. It was Boutet de Monvel [19]

1.3 Summary of the Contents

23

who brought in the operator-algebraic aspect with his calculus in 1971. He introduced a 2 × 2 matrix A of operators, and constructed a relatively small “algebra”, called the Boutet de Monvel algebra, which contains the boundary value problems for elliptic differential operators as well as their parametrices. We will take a close look at Boutet de Monvel’s work. Let Ω be a bounded, domain of Euclidean space Rn with smooth boundary ∂Ω. Without loss of generality, we may assume that Ω is a relatively compact,  without open subset of an n-dimensional, compact smooth manifold M = Ω boundary (see Figure 1.10 below). The manifold M is called the double of Ω.

Ω

M

∂Ω

Fig. 1.10. The bounded domain Ω and the double M

Boutet de Monvel [19] introduced a 2 × 2 matrix A of operators ⎞ ⎛ PΩ + G K C ∞ (Ω) C ∞ (Ω) ⎠ ⎝ : A= −→ T S C ∞ (∂Ω) C ∞ (∂Ω) Here: (1) P is a pseudo-differential operator on the full manifold M and    PΩ u = r+ P (e+ u) = P (e+ u)Ω for all u ∈ C ∞ (Ω), where e+ u is the extension of u by zero to M  u(x) for x ∈ Ω, + e u(x) = 0 for x ∈ M \ Ω, while r+ v is the restriction v|Ω to Ω of a distribution v on M . In view of the pseudo-local property of P , we find that the operator PΩ can be visualized as follows:

24

1 Introduction and Main Results e+

r+

PΩ : C ∞ (Ω) −→ D (M ) −→ D (M ) −→ C ∞ (Ω). P

The crucial requirement is that the symbol of P has the transmission property in order that PΩ maps C ∞ (Ω) into itself. In Section 5.1 we introduce three basic function spaces H, H + and H − (Proposition 5.1). In Section 5.2 we illustrate how the transmission property of the symbol ensures that the associated operator preserves smoothness up to the boundary (Theorems 5.6 and 5.7). (2) S is a pseudo-differential operator on ∂Ω. (3) The potential operator K and trace operator T are generalizations of the potentials and trace operators known from the classical theory of elliptic boundary value problems, respectively. (4) The entry G, a singular Green operator, is an operator which is smoothing in the interior Ω while it acts like a pseudo-differential operator in directions tangential to the boundary ∂Ω. As an example, we may take the difference of two solution operators to (invertible) classical boundary value problems with the same differential part in the interior but different boundary conditions. In Section 5.3 we give typical examples of a potential operator K, a trace operator T and a singular Green operator G (see Examples 5.5 through 5.12). Boutet de Monvel [19] proved that these operator matrices form an algebra in the following sense: Given another element of the calculus, say, ⎞ ⎛  PΩ + G K  C ∞ (Ω) C ∞ (Ω)  ⎠ ⎝ : −→ A = T S C ∞ (∂Ω) C ∞ (∂Ω) the composition A A is again an operator matrix of the type described above. It is worth pointing out here that the product PΩ PΩ does not coincide with (P  P )Ω ; in fact, the difference PΩ PΩ − (P  P )Ω turns out to be a singular Green operator. Section 5.4 is devoted to a brief historical perspective of the Wiener– Hopf technique. It should be emphasized that the Boutet de Monvel calculus is closely related to the classical Wiener–Hopf technique (see [145], [84]) that remains a source of inspiration to Mathematicians, Physicists and Engineers working in many diverse fields, and the areas of application continue to broaden. In Chapter 6 we consider the non-homogeneous general Robin problem ⎧ in Ω, ⎨Au = f   (6.1) ∂u ⎩Bγu := a(x ) + b(x )u = ϕ on ∂Ω ∂ν ∂Ω

1.3 Summary of the Contents

25

under the following two conditions (H.1) and (H.2) (corresponding to conditions (A) and (B) with μ(x ) := a(x ) and γ(x ) := −b(x )): (H.1) a(x ) ≥ 0 and b(x ) ≥ 0 on ∂Ω. (H.2) a(x ) + b(x ) > 0 on ∂Ω. Here ν = −n is the unit outward normal to the boundary ∂Ω. We study the general Robin boundary value problem (6.1) in the framework of Sobolev spaces of Lp type, by using the Lp theory of pseudo-differential operators. The purpose of Section 6.1 is to describe, in terms of pseudodifferential operators, the classical surface and volume potentials arising in boundary value problems for elliptic differential operators. This calculus of pseudo-differential operators is applied to elliptic boundary value problems in Part III. In Section 6.2 we consider the Dirichlet problem in the framework of Sobolev spaces of Lp type. This is a generalization of the classical potential approach to the Dirichlet problem. In Section 6.3 we formulate elliptic boundary value problems in the framework of Sobolev spaces of Lp type. The pseudo-differential operator approach to elliptic boundary value problems can be traced back to the pioneering work of Calder´on [22] in early 1960s ([57], [105]). The idea of our approach is stated as follows. In Section 6.4, we consider the following homogeneous Neumann problem: ⎧ in Ω, ⎨Av =  f (1.18) ∂v  ⎩ =0 on ∂Ω. ∂n ∂Ω The existence and uniqueness theorem for the Neumann problem (1.18) is well established in the framework of Sobolev spaces of Lp type (cf. [4], [50], [77], [133], [146]). More precisely, in the Neumann problem for the differential operator A and its formal adjoint A∗ , we have parabolic condensation of eigenvalues along the negative real axis, as discussed in Agmon [3, pp. 276–277]. We let v := GN f. The operator GN is the Green operator for the homogeneous Neumann problem. Then it follows that a function u(x) is a solution of the general Robin problem (6.1) if and only if the function w(x) = u(x) − v(x) = u(x) − GN f (x) is a solution of the problem ⎧ ⎨Aw = 0    ∂v ⎩Bγw = −Bγv = − a(x ) + b(x )v  = −b(x ) (v|∂Ω ) ∂ν ∂Ω

in Ω, on ∂Ω.

However, we know that every solution w of the homogeneous equation

26

1 Introduction and Main Results

Aw = 0 in Ω can be expressed by means of a single layer potential as follows: w = Pψ. The operator P is the Poisson operator for the Dirichlet problem. Thus, by using the operators GN and P we can reduce the study of the general Robin problem (6.1) to that of the equation T ψ := Bγ (Pψ) = −b(x ) ((GN f )|∂Ω ) = −b(x )γ0 (GN f ) .

(1.19)

This is a modern version of the classical Fredholm integral equation in terms of pseudo-differential operators. It is well known (cf. [26], [57], [61], [73], [93], [105], [133]) that the operator T = BγP = a(x ) γ1 P + b(x ) γ0 P = −a(x ) Π + b(x ), is a pseudo-differential operator of first order on the boundary ∂D. More precisely, Π is called the Dirichlet-to-Neumann operator defined by the formula (cf. [138, p. 134, formula (4.13)])     ∂ ∂  (Pϕ) = (Pϕ) Πϕ = −γ1 P = − for all ϕ ∈ C ∞ (∂Ω), (6.3) ∂ν ∂n ∂Ω ∂Ω and further (cf. [26], [61], [73], [93], [105], [133]) that Π is a classical, elliptic pseudo-differential operator of first order on ∂Ω. The virtue of the boundary equation (1.19) is that there is no difficulty in taking adjoints or transposes after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints or transposes. This allows us to discuss the existence theory more easily. Here it should be emphasized that our reduction approach would break down if we use the Dirichlet problem (Theorem 6.15) as usual, instead of the Neumann problem (Theorem 6.16). The reader might be referred to Iwasaki [66, p. 563, Example]. The study of problem (6.1) can be expressed, in terms of the Boutet de Monvel calculus, in the following matrix formula: ⎞ ⎛ ⎞ ⎛ ⎞⎛ I 0 A 0 GN P ⎠=⎝ ⎠. ⎝ ⎠⎝  0 0 γ(x ) (γ0 GN ) T Bγ 0 In Section 6.5 we study the non-homogeneous general Robin problem  Au = f in Ω, (6.1) Bγu = a(x ) γ1 u + b(x ) γ0 u = ϕ on ∂Ω and the non-homogeneous Neumann problem

1.3 Summary of the Contents

⎧ ⎨Av =  g ∂v  ⎩ =ψ ∂n 

27

in Ω, on ∂Ω

(6.4)

∂D

in the framework of Lp Sobolev spaces from the viewpoint of the Boutet de Monvel calculus. Here   ∂u  ∂u  =− on ∂Ω. γ1 u := ∂ν ∂Ω ∂n ∂Ω Then we derive an index formula of Agranoviˇc–Dynin type for the Neumann problem (6.4) and the general Robin problem (6.1) in the framework of Lp Sobolev spaces (Theorem 6.26). In Section 6.6 we study an intimate relationship between the Dirichlet-toNeumann operator Π and the reflecting diffusion in a bounded, domain Ω of Euclidean space RN with smooth boundary ∂D (Theorems 6.27 and 6.28 and Remark 6.29). In Section 6.7, following Mizohata [82, Chapter 3] and Wloka [146, Section 13] we prove that all the sufficiently large eigenvalues of the Dirichlet problem for the differential operator A lie in the parabolic type region (see assertion (6.52) and Figure 6.3). Our subject proper starts with the third part (Chapters 7 and 9) of this book. Chapter 7 is devoted to the proof of Theorem 1.2. The proof of Theorem 1.2 is flowcharted (Table 7.1). The idea of our proof is stated as follows. First, we reduce the study of the boundary value problem (∗)λ to that of a first-order pseudo-differential operator T (λ) = LP (λ) on the boundary ∂D, just as in Section 6.4. Then we prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate   (1.7)

u 2,p ≤ C(λ) f p + |ϕ|2−1/p,p + u p . More precisely, we construct a parametrix S(λ) for T (λ) in the H¨ ormander class L01,1/2 (∂D) (Lemma 7.2), and apply the Besov-space boundedness theorem (Theorem 4.47) to S(λ) to obtain the desired estimate (1.7) (Lemma 7.1). Chapter 8 and the next Chapter 9 are devoted to the proof of Theorem 1.4. In this chapter we study the operator Ap , and prove a priori estimates for the operator Ap − λI (Theorem 8.3) which will play a fundamental role in the next chapter. In the proof we make good use of Agmon’s method (Proposition 8.4). This is a technique of treating a spectral parameter λ as a second-order, elliptic differential operator of an extra variable and relating the old problem to a new problem with the additional variable. In Chapter 9 we prove Theorem 1.4 (Theorems 9.1 and 9.11). The proof of Theorem 1.4 is flowcharted (Table 9.1). Once again we make use of Agmon’s method in the proof of Theorems 9.1 and 9.11. In particular, Agmon’s method

28

1 Introduction and Main Results

plays an important role in the proof of the surjectivity of the operator Ap − λI (Proposition 9.2). Part IV (Chapters 10 and 11) is devoted to the general study of the maximum principles for second-order, elliptic Waldenfels operators. In Chapter 10, following Bony–Courr`ege–Priouret [15] we prove various maximum principles for second-order, elliptic Waldenfels operators W = P +S which play an essential role throughout the book. In Section 10.1 we give complete characterizations of linear operators W which satisfy the positive maximum principle (PM) closely related to condition (β  ) given in the Hille– Yosida–Ray theorem (Theorem 3.36): ◦

x0 ∈ D, v ∈ C02 (D) and v(x0 ) = sup v(x) ≥ 0 =⇒ W v(x0 ) ≤ 0

(PM)

x∈D

(Theorems 10.1, 10.2 and 10.4). In Section 10.2 we prove the weak and strong maximum principles and Hopf’s boundary point lemma for second-order, elliptic Waldenfels operators W = P + S (Theorems 10.5, 10.7 and Lemma 10.11) that play an important role in Chapters 12 and 13. In Chapter 11, following Bony–Courr`ege–Priouret ([15, Chapter II]) we characterize Ventcel’–L´evy boundary operators T (Theorem 11.3) and general boundary operators Γ = Λ+T (Theorem 11.4) defined on the compact smooth manifold D with boundary ∂D in terms of the positive boundary maximum principle (PMB): x0 ∈ ∂D, u ∈ C 2 (D) and u(x0 ) = max u(x) ≥ 0 =⇒ Γ u(x0 ) ≤ 0. (PMB) x∈D

This chapter is very useful in the study of Markov processes with general Ventcel’ boundary conditions in Chapter 14. The fourth part (Chapters 12 through 14) of this book is devoted to the proofs of main theorems (Theorems 1.5 through 1.11). Chapters 12 and 13 are devoted to the proofs of Theorem 1.5 and Theorem 1.6. In Chapter 12, we prove part (i) of Theorem 1.5. The proof of part (i) of Theorem 1.5 is flowcharted (Table 12.1). Part (i) of Theorem 1.5 follows from Theorem 1.4, by using Sobolev’s imbedding theorems (Theorems 4.2 and 4.6) and a λ-dependent localization argument essentially due to Masuda [78] (Lemma 12.2). Chapter 13 is devoted to the proofs of Theorem 1.6 and part (ii) of Theorem 1.5. This chapter is the heart of the subject. In Section 13.1 we formulate general existence theorems for Feller semigroups in terms of elliptic boundary value problems with spectral parameter (Theorem 13.14). In Section 13.2 we study Feller semigroups with reflecting barrier (Theorem 13.17) and then, by using these Feller semigroups we construct Feller semigroups corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem

1.3 Summary of the Contents

29

13.22). Our proof is based on the generation theorems of Feller semigroups discussed in Section 3.3. In Section 13.3 we prove Theorem 1.6. To do so, we apply part (ii) of Theorem 3.34 to the operator A defined by formula (1.11). The proof of Theorem 1.6 is flowcharted (Table 13.1). Section 13.4 is devoted to the proof of part (ii) of Theorem 1.5. The proof is flowcharted (Table 13.2) In Chapter 14 we study a class of degenerate boundary value problems for second-order elliptic Waldenfels operators W = A + S, and generalize Theorems 1.4 and 1.5 (Theorems 1.8, 1.9, 1.10 and 1.11). In Section 14.1, by using the H¨older space theory of pseudo-differential operators we study the non-homogeneous boundary value problem ⎧ in D, ⎨Au = f   (∗) ∂u ⎩Lu = μ(x ) + γ(x )u = ϕ on ∂D ∂n ∂D in the framework of H¨ older spaces, and prove an existence and uniqueness theorem (Theorem 14.1). More precisely, we prove that if conditions (A) and (B) are satisfied, then the mapping (A, L) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is an algebraic and topological isomorphism for all 0 < θ < 1. In Section 14.2 we prove an existence and uniqueness theorem for the non-homogeneous boundary value problem  W u = (A + S)u = f in D, (∗∗) Lu = ϕ on ∂D. in the framework of H¨ older spaces (Theorem 1.8). The proof of Theorem 1.8 is flowcharted (Table 14.1). The essential point in the proof of Theorem 1.8 is to estimate the integral operator S in terms of H¨ older norms (Lemmas 14.4 and 14.5). We show that if condition (H) is satisfied, then the operator (W, L) = (A, L) + (S, 0) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) may be considered as a perturbation of a compact operator (S, 0) to the operator (A, L) in the framework of H¨older spaces (Lemma 14.6). In this way, the proof of Theorem 1.8 is reduced to the differential operator case (Theorem 14.1). In Section 14.3 we prove Theorem 1.9 (Theorem 14.8). We estimate the integral operator S in terms of Lp norms, and show that S is an Ap -completely continuous operator in the sense of Gohberg–Kre˘ın [51] (Lemmas 14.9 and 14.10). The proof of Theorem 1.9 is flowcharted (Table 14.2).

30

1 Introduction and Main Results

Section 14.4 is devoted to the proof of Theorem 1.10. Theorem 1.10 follows from Theorem 1.9 by using Sobolev’s imbedding theorems and a λ-dependent localization argument. The proof is carried out in a chain of auxiliary lemmas (Lemmas 14.11, 14.12 and 14.15). The proof of Theorem 1.10 is flowcharted (Table 14.3). In Section 14.5, as an application, we construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it “dies” at the time when it reaches the set where the particle is definitely absorbed, generalizing Theorem 1.6 (Theorem 1.11). The proof of Theorem 1.11 is flowcharted (Table 14.4). In this book we have studied mainly Markov transition functions with only informal references to the random variables which actually form the Markov processes themselves. In Chapter 15 we study this neglected side of our subject. Section 15.1 is devoted to a review of the basic definitions and properties of Markov processes. In Section 15.2 we consider when the paths of a Markov process are actually continuous, and prove Theorem 3.19 (Corollary 15.7). In Section 15.3 we give a useful criterion for path-continuity of a Markov process {xt } in terms of the infinitesimal generator A of the associated Feller semigroup {Tt } (Theorem 15.9). Section 15.4 is devoted to the study of three typical examples of multidimensional diffusion processes. More precisely, we prove that (1) the reflecting barrier Brownian motion (Theorem 15.11), (2) the reflecting and absorbing barrier Brownian motion (Theorem 15.14) and (3) the reflecting, absorbing and drifting barrier Brownian motion (Theorem 15.15) are multi-dimensional diffusion processes, namely, they are continuous strong Markov processes. It should be emphasized that these three Brownian motions correspond to (1) the Neumann boundary value problem, (2) the Robin boundary value problem and (3) the oblique derivative boundary value problem for the Laplacian Δ in terms of elliptic boundary value problems, respectively. In the final Chapter 16, as concluding remarks, we give an overview of general results on generation theorems for Feller semigroups proved mainly by the author using the theory of pseudo-differential operators ([57], [105], [106]) and the Calder´ on–Zygmund theory of singular integral operators ([23]). In particular, we generalize Theorem 13.17 to the transversal case (Theorem 16.1) and Theorem 13.22 to the non-transversal case (Theorem 16.2), respectively.

1.4 An Overview of Main Theorems Table 1.4 below gives an overview of analytic and Feller semigroups for the Waldenfels integro-differential operator

1.4 An Overview of Main Theorems

Closed operators

Function spaces

Generated semigroups

Ap (Theorem 1.4)

Lp (D)

ez Ap (analytic)

A (Theorem 1.5)

A (Theorem 1.6)

  C0 D \ M

  C0 D \ M

Lp (D)

Wp (Theorem 1.9)

W (Theorem 1.10)

W=W (Theorem 1.11)

  C0 D \ M

  C0 D \ M

31

ez A (analytic) et A (Feller) ez Wp (analytic) ez W (analytic)

et W = et W (Feller)

Table 1.4. An overview of generation theorems for analytic and Feller semigroups (Theorems 1.4 through 1.11)

W u(x) = Au(x) + Su(x) (1.2) ⎞ ⎛ N N   ∂2u ∂u aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ := ⎝ ∂x ∂x ∂x i j i i,j=1 i=1 ⎛ ⎞  N  ∂u ⎝u(x + z) − u(x) − zj (x)⎠ s(x, z) m(dz) + ∂xj RN \{0} j=1 and the Ventcel’ boundary condition Lu(x ) = μ(x )

∂u  (x ) + γ(x )u(x ). ∂n

(1.6)

32

1 Introduction and Main Results

Both Theorems 1.4 and 1.9 are a generalization of Agranovich–Vishik [6, Theorem 5.1] to the degenerate case. Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Lp theory of pseudo-differential operators which may be considered as a modern version of the classical potential theory. It should be emphasized that pseudo-differential operators provide a constructive tool to deal with existence and smoothness of solutions of partial differential equations. Finally, Table 1.5 below gives a bird’s-eye view of Markov processes and elliptic boundary value problems via the Boutet de Monvel calculus.

Field

Probability (Microscopic approach)

Partial Differential Equations (Mesoscopic approach)

Mathematical subject

Markov processes on the domain

Elliptic boundary value problems

Reduction to the boundary

Markov processes on the boundary

Fredholm integral equations on the boundary

Mathematical theory

Stochastic calculus

Boutet de Monvel calculus

Table 1.5. A bird’s-eye view of Markov processes and elliptic boundary value problems via the Boutet de Monvel calculus

1.5 Notes and Comments This chapter is mainly based on the lecture entitled A mathematical study of diffusion delivered at Mathematisch-Physikalisches Kolloquim, Leibniz Universit¨ at Hannover, Germany, on November 3, 2015. For further study of Ventcel’ boundary value problems for elliptic Waldenfels operators, the reader might be referred to [128, Theorems 1.1 and 1.2].

Part I

Analytic and Feller Semigroups and Markov Processes

2 Analytic Semigroups

This chapter is devoted to a review of standard topics from the theory of analytic semigroups which forms a functional analytic background for the proof of Theorems 1.4 and 1.5.

2.1 Analytic Semigroups via the Cauchy Integral Let E be a Banach space over the real or complex number field, and let A : E → E be a densely defined, closed linear operator with domain D(A). Assume that the operator A satisfies the following two conditions (see Figure 2.1 below): (1) The resolvent set of A contains the region Σω = {λ ∈ C : λ = 0, | arg λ| < π/2 + ω}

for 0 < ω < π/2.

(2.1)

(2) For each ε > 0, there exists a positive constant Mε such that the resolvent −1 R(λ) = (A − λI) satisfies the estimate Mε |λ| for all λ ∈ Σεω = {λ ∈ C : λ = 0, | arg λ| ≤ π/2 + ω − ε}.

R(λ) ≤

Then we let U (t) = −

1 2πi

 eλ t R(λ) dλ = − Γ

1 2πi

 eλ t (A − λI)

−1

dλ.

(2.2)

(2.3)

Γ

Here Γ is a path in the set Σεω consisting of the following three curves (see Figure 2.2 below):   Γ (1) = re−i(π/2+ω−ε) : 1 ≤ r < ∞ ,

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 2

36

2 Analytic Semigroups

Σω 0

Fig. 2.1. The region Σω in condition (1)

  Γ (2) = ei θ : −(π/2 + ω − ε) ≤ θ ≤ π/2 + ω − ε ,   Γ (3) = rei(π/2+ω−ε) : 1 ≤ r < ∞ .

Γ (3) Γ (2) 0

Γ (1) Fig. 2.2. The integral path Γ consisting of Γ (1) , Γ (2) and Γ (3)

It is easy to see that the integral 3

U (t) = −

1  2πi k=1

 eλ t R(λ) dλ Γ (k)

converges in the uniform operator topology of the Banach space L(E, E) for all t > 0, and thus defines a bounded linear operator on E. Here L(E, E) denotes the space of bounded linear operators on E. Furthermore, we have the following proposition: Proposition 2.1. The operators U (t), defined by formula (2.2), form a semigroup on E, that is, they enjoy the semigroup property U (t + s) = U (t) · U (s)

for all t, s > 0.

2.1 Analytic Semigroups via the Cauchy Integral

37

Proof. By Cauchy’s theorem, we may assume that  1 U (s) = − eμ s R(μ) dμ for s > 0. 2πi Γ  Here Γ  is a path obtained from the path Γ by translating each point of Γ to the right by a fixed small positive distance (see Figure 2.3 below). Γ

Γ

0

Fig. 2.3. The integral paths Γ and Γ 

Then we have, by Fubini’s theorem,   1 U (t) · U (s) = eλ t eμ s R(λ) R(μ) dλ dμ (2πi)2 Γ Γ    R(λ) − R(μ) 1 dλ dμ = eλ t eμ s 2 (2πi) Γ Γ  λ−μ     1 1 eμ s = dμ dλ eλ t R(λ) 2πi Γ 2πi Γ  λ − μ     1 eλ t 1 μs dλ dμ. − e R(μ) 2πi Γ  2πi Γ λ − μ We calculate the two terms in the last part. (a) We let eμ s , μ ∈ C. f (μ) = λ−μ Then, by applying the residue theorem we obtain that (see Figure 2.4 below)    f (μ) dμ + f (μ) dμ + f (μ) dμ Γ (1) ∩{|μ|≤r}



π/2+ω−ε

+ −(π/2+ω−ε)

Γ (2)

f (r ei θ )r i ei θ dθ

Γ  (3) ∩{|μ|≤r}

38

2 Analytic Semigroups

|μ| = r •λ 0

Fig. 2.4. The truncated integral path Γ  for the residue theorem

= 2πi Res [f (μ)]μ=λ = −2π i eλ s . However, we have, as r → ∞, 



Γ  (1) ∩{|μ|≤r}

f (μ) dμ −→



Γ  (3) ∩{|μ|≤r}

Γ  (1)

f (μ) dμ,

 f (μ) dμ −→

f (μ) dμ, Γ  (3)

and also    π/2+ω−ε  π/2+ω−ε   iθ  dθ   iθ −rs·sin(ω−ε) λ  −→ 0. ≤ e r i e f re dθ     −(π/2+ω−ε)  − ei θ  −(π/2+ω−ε) r Therefore, we find that 1 2πi

 Γ

eμ s dμ = −eλ s . λ−μ

(b) Similarly, since the path Γ lies to the left of the path Γ  , we find that 1 2πi

 Γ

eλ t dλ = 0. λ−μ

Summing up, we obtain that  1 U (t) · U (s) = − eλ(t+s) R(λ) dλ = U (t + s) for all t, s > 0. 2πi Γ The proof of Proposition 2.1 is complete.  

2.2 Generation Theorem for Analytic Semigroups

39

2.2 Generation Theorem for Analytic Semigroups The next theorem states that the semigroup U (t) can be extended to an analytic semigroup in some sector containing the positive real axis ([123, p. 63, Theorem 3.2]): Theorem 2.2. Assume that the operator A satisfies conditions (2.1) and (2.2). The semigroup U (t), defined by formula (2.3), can be extended to a semigroup U (z) that is analytic in the sector Δω = {z = t + is : z = 0, | arg z| < ω} , and enjoys the following three properties: (a) The operators AU (z) and dU dz (z) are bounded operators on E for each z ∈ Δω , and satisfy the relation dU (z) = AU (z) dz

for all z ∈ Δω .

(2.4)

!0 (ε) and M !1 (ε) (b) For each 0 < ε < ω/2, there exist positive constants M such that !0 (ε)

U (z) ≤ M

AU (z) ≤

!1 (ε) M |z|

for all z ∈ Δ2ε ω ,

(2.5)

for all z ∈ Δ2ε ω ,

(2.6)

where (see Figure 2.5 below) Δ2ε ω = {z ∈ C : z = 0, |arg z| ≤ ω − 2ε} . (c) For each x ∈ E, we have, as z → 0, z ∈ Δ2ε ω , U (z)x −→ x

in E.

Proof. (i) The analyticity of U (z): If λ ∈ Γ (3) and z ∈ Δ2ε ω , that is, if we have the formulas λ = |λ|ei θ ,

θ = π/2 + ω − ε,

z = |z|e ,

|ϕ| ≤ ω − 2ε,

iϕ

then it follows that λz = |λ| |z|ei(θ+ϕ) , with

Note that

π 3π π + ε ≤ θ + ϕ ≤ + 2ω − 3ε < − 3ε. 2 2 2

40

2 Analytic Semigroups

Δ2ε ω 0

Fig. 2.5. The sector Δ2ε ω in Theorem 2.2

cos(θ + ϕ) ≤ cos(π/2 + ε) = − sin ε. Hence we have the inequality |eλ z | ≤ e−|λ| |z| sin ε

for all λ ∈ Γ (3) and z ∈ Δ2ε ω .

(2.7)

for all λ ∈ Γ (1) and z ∈ Δ2ε ω .

(2.8)

Similarly, we have the inequality |eλ z | ≤ e−|λ| |z| sin ε For each small ε > 0, we let Kωε = Δ2ε ω ∩ {z ∈ C : |z| ≥ ε} = {z ∈ C : |z| ≥ ε, |arg z| ≤ ω − 2ε} . Then, by combining estimates (2.2), (2.7) and (2.8) we obtain that  λz  e R(λ) ≤ Mε e−ε sin ε·|λ| |λ|

for all λ ∈ Γ (1) ∪ Γ (3) and z ∈ Kωε .

On the other hand, we have the estimate  λz  e R(λ) ≤ Mε e|z| for all λ ∈ Γ (2) and z ∈ Kωε .

(2.9)

(2.10)

Therefore, we find that the integral U (z) = −

1 2πi



3

eλ z R(λ) dλ = − Γ

1  2πi

k=1

 eλ z R(λ) dλ

(2.11)

Γ (k)

converges in the Banach space L(E, E), uniformly in z ∈ Kωε , for every ε > 0. This proves that the operator U (z) is analytic in the domain " Kωε . Δω = ε>0

By the analyticity of U (z), it follows that the operators U (z) also enjoy the semigroup property

2.2 Generation Theorem for Analytic Semigroups

U (z + w) = U (z) · U (w)

41

for all z, w ∈ Δω .

(ii) We prove that the operators U (z) enjoy properties (a), (b) and (c). (b) First, by using Cauchy’s theorem we obtain that   1 1 U (z) = − eλ z R(λ) dλ = − eλ z R(λ) dλ, 2πi Γ 2πi Γ|z| where Γ|z| is a path consisting of the following three curves (see Figure 2.6 below):   1 (1) −i(π/2+ω−ε) Γ|z| = re ≤r 0 such that the resolvent R(λ) = (A − λI) satisfies the estimate

R(λ) ≤

M 1 + |λ|

for all λ ∈ Σ.

(2.16)

Σ

0

Fig. 2.8. The region Σ in condition (1)

We define the semigroup U (t) by the formula (by replacing formula (2.3)) U (t) = −

1 2πi

 eλ t R(λ) dλ = − Γ

1 2πi



eλ t (A − λI)−1 dλ.

(2.3 )

Γ

Here the path Γ runs in the set Σ from ∞e−iω to ∞eiω , avoiding the positive real axis and the origin (see Figure 2.9 below). Then we can prove the following remark:

46

2 Analytic Semigroups

Γ

0

Fig. 2.9. The integral path Γ in formula (2.3 )

Remark 2.3. The estimates (2.5) and (2.6) can be replaced as follows: !0 (ε) e−δ·Re z

U (z) ≤ M

for all z ∈ Δ2ε ω ,

(2.17)

!1 (ε) M e−δ·Re z

AU (z) ≤ |z|

for all z ∈ Δ2ε ω ,

(2.18)

with some constant δ > 0. Proof. Take a real number δ such that 0 0, and we write lim f (x) = 0.

x→∂

We define a subspace C0 (X) of C(X) as follows:

52

3 Markov Processes and Feller Semigroups

 C0 (X) =

 f ∈ C(X) : lim f (x) = 0 . x→∂

It is easy to see that C0 (X) is a Banach space with the supremum (maximum) norm

f ∞ = sup |f (x)|. x∈X

The next proposition asserts that C0 (X) is the uniform closure of Cc (X) (see [42, Chapter 4, Proposition 4.35]): Proposition 3.4. Let (X, ρ) be a locally compact metric space. The space C0 (X) is the closure of Cc (X) in the topology of uniform convergence. Proof. Assume that {fn } is a sequence in Cc (X) which converges uniformly to some function f ∈ C(X). Then, for any given ε > 0 there exists a number n ∈ N such that

f − fn ∞ < ε. Hence we have the assertion |f (x)| < ε

if x ∈ X \ supp fn .

This proves that the set {x ∈ X : |f (x)| ≥ ε} is compact for every ε > 0, so that f ∈ C0 (X). Conversely, if f ∈ C0 (X), we let   1 Kn = x ∈ X : |f (x)| ≥ for n ∈ N. n Since Kn is compact, by applying Urysohn’s lemma (Lemma 3.1) we can find a function gn ∈ Cc (X) such that 0 ≤ gn ≤ 1 and gn = 1 on Kn . Then it follows that fn = gn f ∈ Cc (X) and that

f − fn ∞ = (1 − gn )f ∞ ≤

1 n

for all n ∈ N.

This proves that {fn } converges uniformly to f ∈ C(X). The proof of Proposition 3.4 is complete.   3.1.2 Space of Signed Measures Let (X, M) be a measurable space. A real-valued function μ defined on the σ-algebra M is called a signed measure or real measure if it is countably additive, that is, % $∞ ∞   Ai = μ (Ai ) μ i=1

i=1

for any disjoint countable collection {Ai }∞ i=1 of &members of M. It should be noticed that every rearrangement of the series i μ (Ai ) also converges, since

3.1 Continuous Functions and Measures

53

&∞ the disjoint union i=1 Ai is not changed if the subscripts are permuted. A signed measure takes its values in (−∞, ∞), but a non-negative measure may take ∞; hence the non-negative measures do not form a subclass of the signed measures. If μ and λ are signed measures on M, we define the sum μ + λ and the scalar multiple α μ as follows: (μ + λ) (A) = μ(A) + λ(A) for all A ∈ M, (α μ) (A) = α μ(A) for all α ∈ R and A ∈ M. Then it is clear that μ + λ and α μ are signed measures. If μ is a signed measure, we define a function |μ| on M by the formula '  n  |μ (Ai ) | for all A ∈ M, |μ|(A) = sup i=1

where the supremum is taken over all finite partitions {Ai } of A into members of M. Then the function |μ| is a finite non-negative measure on M. The measure |μ| is called the total variation measure of μ, and the quantity |μ|(X) is called the total variation of μ. We remark that |μ(A)| ≤ |μ| (A) ≤ |μ| (X) for all A ∈ M.

(3.1)

Furthermore, we can verify that the quantity |μ|(X) satisfies the axioms of a norm. Thus the totality of signed measures on M is a normed linear space by the norm μ := |μ|(X). If we define two functions μ+ and μ− on M by the formulas 1 (|μ| + μ) , 2 1 μ− = (|μ| − μ) , 2

μ+ =

then it follows from inequalities (3.31) that both μ+ and μ− are finite nonnegative measures on M. It should be emphasized that the measures μ+ and μ− are the positive and negative variation measures of μ, respectively. We also have the Jordan decomposition of μ: μ = μ+ − μ− . 3.1.3 The Riesz–Markov Representation Theorem First, we characterize the non-negative linear functionals on C(K). We show that non-negative linear functionals on the spaces of continuous functions are given by integration against Radon measures. This fact constitutes an essential link between measure theory and functional analysis, providing a powerful tool for constructing measures.

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3 Markov Processes and Feller Semigroups

Let (X, ρ) be a locally compact metric space and K a compact subset of X, and Cc (X) = {f ∈ C(X) : supp f is compact} . C0 (X) = {f ∈ C(X) : f vanishes at infinity} . First, we characterize the non-negative linear functionals on Cc (X). A linear functional I on Cc (X) is said to be non-negative if I(f ) ≥ 0 whenever f ≥ 0, that is, if it satisfies the condition f ∈ Cc (X), f (x) ≥ 0 on X =⇒ I(f ) ≥ 0. Then we can prove that non-negativity implies a rather strong continuity property (see [42, Chapter 7, Proposition 7.1]): Proposition 3.5. If I is a non-negative linear functional on Cc (X), then, for every compact set K ⊂ X there exists a positive constant CK such that |I(f )| ≤ CK f ∞

for all f ∈ Cc (X) with supp f ⊂ K.

Here

f ∞ = sup |f (x)| . x∈X

If μ is Borel measure on X such that μ(K) < ∞ for every compact set K ⊂ X, then it follows that Cc (X) ⊂ L1 (X, μ). Hence the map

 Iμ : f −→

f (x) dμ(x) X

is a non-negative linear functional on the space Cc (X). The purpose of this subsection is to prove that every non-negative linear functional on Cc (X) arises in this fashion. In doing this, we impose some additional conditions on μ, subject to which μ is uniquely determined. A Radon measure on X is a Borel measure that is finite on all compact sets in X, and is outer regular on all Borel sets in X and inner regular on all open sets in X. If U is an open set in X and f ∈ Cc (X), then we write f ≺U to mean that (see Figure 3.3 below)  0 ≤ f (x) ≤ 1 supp f ⊂ U.

on X,

3.1 Continuous Functions and Measures

55

f

U f ≺U

Fig. 3.3. The notation f ≺ U

On the other hand, if K is a subset of X, we let  1 if x ∈ K, χK (x) = 0 if x ∈ X \ K, and we write f ≥ χK to mean that (see Figure 3.4 below) f (x) ≥ χK (x)

on X.

f χK

K f ≥ χK

Fig. 3.4. The notation f ≥ χK

The next theorem asserts that non-negative linear functionals on the space Cc (X) are given by integration against Radon measures (see [42, Chapter 7, Theorem 7.2]): Theorem 3.6 (the Riesz representation theorem). Let (X, ρ) be a locally compact metric space. If I is a non-negative linear functional on the space Cc (X), then there is a unique Radon measure μ on X such that  I(f ) = f (x) dμ(x) for all f ∈ Cc (X). (3.2) X

56

3 Markov Processes and Feller Semigroups

Furthermore, the Radon measure μ enjoys the following two properties (3.3) and (3.4): • μ(U ) = sup {I(f ) : f ∈ Cc (X), f ≺ U } for every open set U ⊂ X.

(3.3)

• μ(K) = inf {I(f ) : f ∈ Cc (X), f ≥ χK }

(3.4)

for every compact set K ⊂ X. Proof. The proof is divided into two steps. Step 1: First, we prove the uniqueness of a Radon measure. More precisely, we show that a Radon measure μ is determined by I on all Borel subsets of X. Assume that μ is a Radon measure such that  I(f ) = f (x) dμ for all f ∈ Cc (X). X

Let U be an arbitrary open subset of X. Then we have, for every function f ≺ U,   f (x) dμ ≤ dμ = μ(U ). I(f ) = X

U

On the other hand, if K is a compact subset of U , then it follows from an application of Urysohn’s lemma (Lemma 3.1) that there exists a function f ∈ Cc (X) such that f ≺ U and f = 1 on K. Hence we have the inequality   χK (x) dμ ≤ f (x) dμ = I(f ). μ(K) = X

X

However, since μ is inner regular, we obtain that μ(U ) = sup {μ(K) : K ⊂ U, K is compact} ≤ sup {I(f ) : f ∈ Cc (X), f ≺ U } ≤ μ(U ). Therefore, we have, for every open set U ⊂ X, μ(U ) = sup {I(f ) : f ∈ Cc (X), f ≺ U } . This proves that the Radon measure μ is determined by I on open subsets U of X, and hence by I on all Borel subsets of X, since it is outer regular on all Borel sets. Step 2: The proof of the uniqueness suggests how to construct a Radon measure μ. More precisely, we begin by defining μ(U ) for an arbitrary open set U ⊂ X by the formula μ(U ) = sup {I(f ) : f ∈ Cc (X), f ≺ U } , and then define μ∗ (E) for an arbitrary set E ⊂ X by the formula

3.1 Continuous Functions and Measures

57

μ∗ (E) = inf {μ(U ) : U ⊃ X, U is open} . It should be noticed that μ∗ (U ) = μ(U ) if U is open, since we have μ(U ) ≤ μ(V ) for U ⊂ V . The idea of the proof may be explained as follows: (i) First, we prove that μ∗ is an outer measure: μ∗ (∅) = 0. μ∗ (E) ≤ μ∗ (F ) if E ⊂ F . ⎛ ⎞ ∞ ∞ "  μ∗ ⎝ Ej ⎠ ≤ μ∗ (Ej ). j=1

j=1

(ii) Secondly, we show that every open subset U of X is μ∗ -measurable: μ∗ (E) = μ∗ (E ∩ U ) + μ∗ (E \ U ) for all E ⊂ X such that μ∗ (E) < ∞. It follows from an application of Carath´eodory’s theorem ([42, Theorem 1.11]) that every Borel set is μ∗ -measurable and further that the restriction μ of μ∗ to the σ-algebra BX is a Borel measure. It should be emphasized that μ∗ (U ) = μ(U ) if U is open and that the measure μ is outer regular and satisfies condition (3.3). (iii) Thirdly, we prove that the measure μ satisfies the condition (3.3). This implies that μ is finite on compact subsets of X and is inner regular on open subsets U of X: μ(U ) = sup {μ(K) : K ⊂ U, K is compact} . Indeed, if U is open and if α is an arbitrary number satisfying the condition α < μ(U ), then we can choose a function f ∈ Cc (X) such that f ≺ U and that I(f ) > α. We let K = supp f. If g is a function in Cc (X) satisfying the condition g ≥ χK , then it follows that g−f ≥0 so that, by the positivity of I, I(g) ≥ I(f ) > α. However, we have, by formula (3.34), μ(K) > α. This proves that μ is inner regular on open sets, since α is an arbitrary number satisfying the condition α < μ(U ).

58

3 Markov Processes and Feller Semigroups

(iv) Finally, we prove formula (3.2). Proof of Assertion (i): It suffices to show that if {Uj } is a sequence of open sets in X and if U = ∪∞ j=1 Uj , then we have the inequality μ(U ) ≤

∞ 

μ (Uj ) .

j=1

Indeed, it follows from this inequality that we have, for any subset E ⊂ X, ⎧ ⎫ ∞ ∞ ⎨ ⎬ " μ∗ (E) = inf μ (Uj ) : E ⊂ Uj , Uj is open , ⎩ ⎭ j=1

j=1

and further ([42, Proposition 1.10]) that the expression of the right-hand side defines an outer measure. If U = ∪∞ j=1 Uj and if f ∈ Cc (X) such that f ≺ U , then we let K = supp f. Since K is compact, it follows that, for some finite n, K⊂

n "

Uj .

j=1

Moreover, we can find functions g1 , g2 , . . ., gn ∈ Cc (X) such that gj ≺ Uj and &n g = 1 on K (a partition of unity subordinate to the covering {Uj }). j=1 j However, since we have the formula f=

n 

f gj ,

f gj ≺ Uj ,

j=1

we obtain that, for any function f ≺ U , I(f ) =

n  j=1

I (f gj ) ≤

n 

μ (Uj ) ≤

j=1

∞ 

μ (Uj ) ,

j=1

so that μ(U ) = sup {I(f ) : f ∈ Cc (X), f ≺ U } ≤

∞ 

μ (Uj ) .

j=1

Proof of Assertion (ii): It suffices to show that μ∗ (E) ≥ μ∗ (E ∩ U ) + μ∗ (E \ U )

(3.5) ∗

for all E ⊂ X such that μ (E) < ∞.

3.1 Continuous Functions and Measures

59

First, we consider the case where E is open. Then, for any given ε > 0 we can find a function f ∈ Cc (X) such that  f ≺ E ∩ U, I(f ) > μ(E ∩ U ) − ε. Moreover, since the set E \ supp f is also open, we can find a function g ∈ Cc (X) such that  g ≺ E \ supp f, I(g) > μ(E \ supp f ) − ε. However, we have the assertions f + g ≺ E,

supp f ⊂ U,

and so μ(E) ≥ I(f ) + I(g) > μ (E ∩ U ) + μ (E \ supp f ) − 2ε ≥ μ∗ (E ∩ U ) + μ∗ (E \ U ) − 2ε. Therefore, by letting ε ↓ 0 in this inequality we obtain the desired inequality (3.5). Secondly, we consider the general case where μ∗ (E) < ∞. Then, for any given ε > 0 we can find an open subset V ⊃ E such that μ(V ) < μ∗ (E) + ε. Hence it follows that μ∗ (E) + ε > μ(V ) ≥ μ∗ (V ∩ U ) + μ∗ (V \ U ) ≥ μ∗ (E ∩ U ) + μ∗ (E \ U ) . Therefore, by letting ε ↓ 0 in this inequality we obtain the desired inequality (3.5). Proof of Assertion (iii): Let K be an arbitrary compact subset of X, and let f ∈ Cc (X) such that f ≥ χK . If ε is an arbitrary positive number, we define an open set Uε as follows: Uε = {x ∈ X : f (x) > 1 − ε} . Then it follows that we have, for any function g ≺ Uε , 1 f − g ≥ 0, 1−ε and so, by the positivity of I, I(g) ≤

1 I(f ). 1−ε

60

3 Markov Processes and Feller Semigroups

Hence we have the inequality μ(K) ≤ μ(Uε ) = sup {I(g) : g ∈ Cc (X), g ≺ Uε } ≤

1 I(f ). 1−ε

Therefore, by letting ε ↓ 0 in this inequality we obtain that μ(K) ≤ I(f ). This proves that we have, for every compact set K ⊂ X, μ(K) ≤ inf {I(f ) : f ∈ Cc (X), f ≥ χK } .

(3.6)

On the other hand, for any open set U ⊃ K, by using Urysohn’s lemma (Lemma 3.1) we can find a function h ∈ Cc (X) such that h ≥ χK and h ≺ U . Hence we have the inequality I(h) ≤ μ(U ) = sup {I(f ) : f ∈ Cc (X), f ≺ U } . However, since μ is outer regular on K, it follows that μ(K) = inf {μ(U ) : U ⊃ K, U is open} . Hence we have proved that I(h) ≤ μ(K). This proves that inf {I(f ) : f ∈ Cc (X), f ≥ χK } ≤ I(h) ≤ μ(K).

(3.7)

Therefore, the desired formula (3.4) follows by combining inequalities (3.6) and (3.7). Proof of Assertion (iv): To do this, we have only to show that  f (x)dμ for all f ∈ Cc (X, [0, 1]). I(f ) = X

Indeed, it suffices to note that the space Cc (X) is the linear span of functions in the space Cc (X, [0, 1]). For any positive integer N ∈ N, we let   j Kj = x ∈ X : f (x) ≥ for 1 ≤ j ≤ N , N and K0 = supp f. Moreover, we define functions f1 , f2 , . . ., fN ∈ Cc (X, [0, 1]) by the formulas     1 j−1 ,0 , fj (x) = min max f (x) − for 1 ≤ j ≤ N . N N

3.1 Continuous Functions and Measures

Here it should be noticed that ⎧ ⎪ ⎨0 fj (x) = f (x) − ⎪ ⎩1

j−1 N

N

Then it follows that so that

61

if x ∈ / Kj−1 , if x ∈ Kj−1 \ Kj , if x ∈ Kj .

1 1 χKj ≤ fj ≤ χKj−1 , N N

  1 1 μ (Kj ) = χKj (x) dμ ≤ fj (x) dμ N N X X  1 1 ≤ χKj−1 (x) dμ = μ (Kj−1 ) . N X N

(3.8)

Also, if U is an open set containing Kj−1 , then we have the condition N fj ≺ U, and the inequality μ(U ) . N Hence, by formula (3.4) and outer regularity of μ it follows that   1 1 μ (Kj ) = inf I(f ) : f ∈ Cc (X), f ≥ χKj ≤ I (fj ) (3.9) N N 1 1 inf {μ(U ) : U ⊃ Kj−1 , U is open} = μ (Kj−1 ) . ≤ N N However, since we have the formula I (fj ) ≤

f=

N 

fj ,

j=1

it follows from inequalities (3.8) and (3.9) that  N N  N −1  1  1  μ (Kj ) ≤ fj (x) dμ = f (x)dμ ≤ μ (Kj ) , N j=1 N j=0 X j=1 X N N N −1  1  1  μ (Kj ) ≤ I (fj ) = I(f ) ≤ μ (Kj ) . N j=1 N j=0 j=1

Hence we have the inequalities     μ (K0 ) − μ (KN )  μ (supp f ) I(f ) − ≤ f (x) dμ ≤  N N X

for all N ∈ N.

Therefore, by letting N → ∞ in this inequality we obtain the desired formula (3.2), since μ (supp f ) < ∞. Now the proof of Theorem 3.6 is complete.  

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3 Markov Processes and Feller Semigroups

We recall (Proposition 3.4) that C0 (X) is the uniform closure of Cc (X). Hence we find that if μ is a Radon measure on X, then the linear functional  Iμ : f −→ f (x) dμ(x) X

extends continuously to C0 (X) if and only if it is bounded with respect to the uniform norm. This happens only when μ(X) = sup {I(f ) : f ∈ Cc (X), 0 ≤ f ≤ 1 on X} < ∞, in which case μ(X) is the operator norm I of F . Therefore, we have the following locally compact version of the Riesz– Markov representation theorem (see [42, Chapter 7, Theorem 7.17 and Corollary 7.18]): Theorem 3.7 (the Riesz–Markov representation theorem). Let (X, ρ) be a locally compact metric space. If F is a non-negative linear functional on the space C0 (X), then there exists a unique Radon measure μ on X such that  F (f ) = f (x) dμ(x) for all f ∈ C0 (X), X

and we have the formula   f (x) dμ(x) : f ∈ C0 (X), 0 ≤ f ≤ 1 on X = F . μ(X) = sup X

Corollary 3.8. Let (K, ρ) be a compact metric space. Then we have the following two assertions (i) and (ii): (i) To each non-negative linear functional T on C(K), there corresponds a unique Radon measure μ on K such that  T (f ) = f (x) dμ(x) for all f ∈ C(K), (3.10) K

and we have the formula

T = μ(K).

(3.11)

(ii) Conversely, every finite Radon measure μ on K defines a non-negative linear functional T on C(K) through formula (3.10), and relation (3.11) holds true. Remark 3.9. It is easy to see that every open set in a compact metric space is a σ-compact. Thus we find that every finite Radon measure μ is regular.

3.1 Continuous Functions and Measures

63

Now we can characterize the space of all bounded linear functionals on C(K), that is, the dual space C(K) of C(K). Remark that the dual space C(K) is a Banach space with the operator norm

T =

sup |T f |.

f ∈C(K) f ≤1

The next theorem is a compact version of the Riesz–Markov representation theorem: Theorem 3.10 (Riesz–Markov). Let (K, ρ) be a compact metric space. Then we have the following two assertions (i) and (ii): (i) To each T ∈ C(K) , there corresponds a unique real Borel measure μ on K such that formula (3.40) holds true for all f ∈ C(K), and we have the formula

T = the total variation |μ|(K) of μ. (3.12) (ii) Conversely, every real Borel measure μ on K defines a bounded linear functional T ∈ C(K) through formula (3.40), and relation (3.42) holds true. Remark 3.11. The positive and negative variation measures μ+ , μ− of a real Borel measure μ are both regular. We recall that the space of all real Borel measures μ on K is a normed linear space by the norm

μ = the total variation |μ|(K) of μ.

(3.13)

Therefore, we can restate Theorem 3.10 as follows: Theorem 3.12. The dual space C(K) of C(K) can be identified with the space of all real Borel measures on K normed by formula (3.43). 3.1.4 Weak Convergence of Measures Let K be a compact metric space and let C(K) be the Banach space of realvalued continuous functions on K with the supremum (maximum) norm

f ∞ = sup |f (x)|. x∈K

A sequence {μn }∞ n=1 of real Borel measures on K is said to converge weakly to a real Borel measure μ on K if it satisfies the condition   lim f (x) dμn (x) = f (x) dμ(x) for every f ∈ C(K). (3.14) n→∞

K

K

64

3 Markov Processes and Feller Semigroups

Theorem 3.12 asserts that the space of all real Borel measures on K normed by formula (3.13) can be identified with the strong dual space C(K) of C(K). Thus the weak convergence (3.14) of real Borel measures is just the weak* convergence of C(K) . Two more results are important when studying the weak convergence of measures (see [20, Chapter 3, Corollary 3.30], [44, Chapter 4, Theorem 4.12.3], [100, Chapter 8, Theorem 8.13], [147, Chapter V, Section 1, Theorem 10]): Theorem 3.13. The Banach space C(K) is separable, that is, it contains a countable, dense subset. Corollary 3.14. Let X be a separable Banach space. Every bounded sequence ∗ {fn }∞ has a subsequence which converges n=1 in the strong dual space X ∗ weakly* to an element f of X . The next theorem is one of the fundamental theorems in measure theory: Theorem 3.15. Every sequence {μn }∞ n=1 of real Borel measures on K satisfying the condition (3.15) sup |μn | (K) < +∞ n≥1

has a subsequence which converges weakly to a real Borel measure μ on K. Furthermore, if the measures μn are all non-negative, then the measure μ is also non-negative. Proof. By virtue of Theorem 3.13, we can apply Corollary 3.14 with X := C(K) to obtain the first assertion, since condition (3.15) implies the boundedness of the Borel measures μn . The second assertion is an immediate consequence of the first assertion of Corollary 3.8. The proof of Theorem 3.15 is complete.  

3.2 Elements of Markov Processes This section provides a brief description of basic definitions and results about Markov processes. 3.2.1 Definition of Markov Processes Let (Ω, F ) be a measurable space. A non-negative measure P on F is called a probability measure if P (Ω) = 1. The triple (Ω, F , P ) is called a probability space. The elements of Ω are known as sample points, those of F as events and the values P (A), A ∈ F, are their probabilities. An extended real-valued, F -measurable function X on Ω is called a random variable. The integral

3.2 Elements of Markov Processes

65

 X(ω) P (dω) Ω

(if it exists) is called the expectation of X, and is denoted by E(X). We begin with a review of conditional probabilities and conditional expectations (see [122, Chapter 2, Section 2.6]). Let G be a σ-algebra contained in F . If X is an integrable random variable, then the conditional expectation of X for given G is any random variable Y which satisfies the following two conditions (CE1) and (CE2): (CE1) +The function Y is+G-measurable. (CE2) A Y (ω) P (dω) = A X(ω) P (dω) for all A ∈ G. We recall that conditions (CE1) and (CE2) determine Y up to a set in G of measure zero. We shall write Y = E (X | G) . Hence we have, for all A ∈ G,   X(ω) P (dω) = E(X | G) P (dω). A

A

When X is the characteristic function χB of a set B ∈ F , we shall write P (B | G) = E (χB | G) . The function P (B | G) is called the conditional probability of B for given G. This function can also be characterized as follows: (CP1) The function P (B | G) is G-measurable. (CP2) P (A ∩ B) = E (P (B | G); A) for every A ∈ G. Namely, we have, for every A ∈ G,  P (A ∩ B) = P (B | G) (ω) P (dω). A

It should be emphasized that the function P (B | G) is determined up to a set in G of P -measure zero, that is, it is an equivalence class of G-measurable functions on Ω with respect to the measure P . Markov processes are an abstraction of the idea of Brownian motion. Let K be a locally compact, separable metric space and let B be the σ-algebra of all Borel sets in K, that is, the smallest σ-algebra containing all open sets in K. Let (Ω, F , P ) be a probability space. A function X(ω) defined on Ω taking values in K is called a random variable if it satisfies the condition X −1 (E) = {ω ∈ Ω : X(ω) ∈ E} ∈ F

for all E ∈ B.

We express this by saying that X is F /B-measurable. A family {xt }t≥0 of random variables is called a stochastic process, and it may be thought of as the motion in time of a physical particle. The space K is called the state space and Ω the sample space. For a fixed ω ∈ Ω, the function xt (ω), t ≥ 0, defines in the state space K a trajectory or path of the process corresponding to the sample point ω.

66

3 Markov Processes and Feller Semigroups

In this generality the notion of a stochastic process is of course not so interesting. The most important class of stochastic processes is the class of Markov processes which is characterized by the Markov property. Intuitively, this is the principle of the lack of any “memory” in the system. More precisely, (temporally homogeneous) Markov property is that the prediction of subsequent motion of a particle, knowing its position at time t, depends neither on the value of t nor on what has been observed during the time interval [0, t); that is, a particle “starts afresh”. Now we introduce a class of Markov processes which we will deal with in this book (see [34, Chapter III, Section 1], [14], [94]). Definition 3.16. Assume that we are given the following: (1) A locally compact, separable metric space K and the σ-algebra B of all Borel sets in K. A point ∂ is adjoined to K as the point at infinity if K is not compact, and as an isolated point if K is compact (see Figure 3.5 below). We let K∂ = K ∪ {∂}, B∂ = the σ-algebra in K∂ generated by B.

K

K∂

∂

Fig. 3.5. The compactification K∂ of K

(2) The space Ω of all mappings ω : [0, ∞] → K∂ such that ω(∞) = ∂ and that if ω(t) = ∂ then ω(s) = ∂ for all s ≥ t. Let ω∂ be the constant map ω∂ (t) = ∂ for all t ∈ [0, ∞]. (3) For each t ∈ [0, ∞], the coordinate map xt defined by xt (ω) = ω(t) for all ω ∈ Ω. (4) For each t ∈ [0, ∞], a mapping ϕt : Ω → Ω defined by (ϕt ω)(s) = ω(t + s) for all ω ∈ Ω. Note that ϕ∞ ω = ω∂ and xt ◦ϕs = xt+s for all t, s ∈ [0, ∞]. (5) A σ-algebra F in Ω and an increasing family {Ft }0≤t≤∞ of sub-σ-algebras of F . (6) For each x ∈ K∂ , a probability measure Px on (Ω, F ).

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67

We say that these elements define a (temporally homogeneous) Markov process X = (xt , F , Ft , Px ) if the following four conditions (i) through (iv) are satisfied: (i) For each 0 ≤ t < ∞, the function xt is Ft /B∂ -measurable, that is, {xt ∈ E} = {ω ∈ Ω : xt (ω) ∈ E} ∈ Ft

for all E ∈ B∂ .

(ii) For all 0 ≤ t < ∞ and E ∈ B, the function pt (x, E) = Px {xt ∈ E}

(3.16)

is a Borel measurable function of x ∈ K. (iii) Px {ω ∈ Ω : x0 (ω) = x} = 1 for each x ∈ K∂ . (iv) For all t, h ∈ [0, ∞], x ∈ K∂ and E ∈ B∂ , we have the formula Px {xt+h ∈ E | Ft } = ph (xt , E)

a. e.,

or equivalently,  Px (A ∩ {xt+h ∈ E}) =

ph (xt (ω), E) dPx (ω)

for all A ∈ Ft .

A

E

t

x

Fig. 3.6. The transition probability pt (x, E)

Here is an intuitive way of thinking about the above definition of a Markov process. The sub-σ-algebra Ft may be interpreted as the collection of events which are observed during the time interval [0, t]. The value Px (A), A ∈ F , may be interpreted as the probability of the event A under the condition that a particle starts at position x; hence the value pt (x, E) expresses the transition probability that a particle starting at position x will be found in the set E at time t (see Figure 3.6 above). The function pt (x, ·) is called the transition function of the process X . The transition function pt (x, ·) specifies the probability structure of the process. The intuitive meaning of the crucial condition (iv) is that the future behavior of a particle, knowing its history up

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3 Markov Processes and Feller Semigroups

to time t, is the same as the behavior of a particle starting at xt (ω), that is, a particle starts afresh. A Markovian particle moves in the space K until it “dies” or “disappear” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point or cemetery. With this interpretation in mind, we let ζ(ω) = inf {t ∈ [0, ∞] : xt (ω) = ∂} . The random variable ζ is called the lifetime of the process X . The process X is said to be conservative if it satisfies the condition Px {ζ = ∞} = 1

for all x ∈ K.

3.2.2 Markov Processes and Markov Transition Functions In the first works devoted to Markov processes, the most fundamental was A. N. Kolmogorov’s work ([70]) where the general concept of a Markov transition function was introduced for the first time and an analytic method of describing Markov transition functions was proposed. From the point of view of analysis, the transition function is something more convenient than the Markov process itself. In fact, it can be shown that the transition functions of Markov processes generate solutions of certain parabolic partial differential equations such as the classical diffusion equation; and, conversely, these differential equations can be used to construct and study the transition functions and the Markov processes themselves. In the 1950s, the theory of Markov processes entered a new period of intensive development. We can associate with each transition function in a natural way a family of bounded linear operators acting on the space of continuous functions on the state space, and the Markov property implies that this family forms a semigroup. The Hille–Yosida theory of semigroups in functional analysis made possible further progress in the study of Markov processes. Our first job is thus to give the precise definition of a transition function adapted to the theory of semigroups (see [34, Chapter III, Section 2]): Definition 3.17. Let (K, ρ) be a locally compact, separable metric space and let B be the σ-algebra of all Borel sets in K. A function pt (x, E), defined for all t ≥ 0, x ∈ K and E ∈ B, is called a (temporally homogeneous) Markov transition function on K if it satisfies the following four conditions (a) through (d): (a) pt (x, ·) is a non-negative measure on B and pt (x, K) ≤ 1 for all t ≥ 0 and x ∈ K. (b) pt (·, E) is a Borel measurable function for all t ≥ 0 and E ∈ B. (c) p0 (x, {x}) = 1 for all x ∈ K. (d) (The Chapman–Kolmogorov equation) For all t, s ≥ 0, x ∈ K and E ∈ B, we have the equation

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69

 pt+s (x, E) =

pt (x, dy) ps (y, E).

(3.17)

K

E

t+s

t

0

y

x K

Fig. 3.7. An intuitive meaning of the Chapman–Kolmogorov equation (3.17)

Here is an intuitive way of thinking about the above definition of a Markov transition function. The value pt (x, E) expresses the transition probability that a physical particle starting at position x will be found in the set E at time t. The Chapman–Kolmogorov equation (3.17) expresses the idea that a transition from the position x to the set E in time t + s is composed of a transition from x to some position y in time t, followed by a transition from y to the set E in the remaining time s; the latter transition has probability ps (y, E) which depends only on y (see Figure 3.7 above). Thus a particle “starts afresh”; this property is called the Markov property. The Chapman–Kolmogorov equation (3.17) asserts that pt (x, K) is monotonically increasing as t ↓ 0, so that the limit p+0 (x, K) = lim pt (x, K) t↓0

exists. A transition function pt (x, ·) is said to be normal if it satisfies the condition p+0 (x, K) = 1

for all x ∈ K.

The next theorem, due to Dynkin [33, Chapter 4, Section 2] (or [34, p. 85, Theorem 3.2]), justifies the definition of a transition function, and hence it will be fundamental for our further study of Markov processes: Theorem 3.18 (Dynkin). For every Markov process, the function pt (x, ·), defined by formula (3.16), is a Markov transition function. Conversely, every normal Markov transition function corresponds to some Markov process.

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3 Markov Processes and Feller Semigroups

Here are some important examples of normal transition functions on the line R = (−∞, ∞): Example 3.1 (Uniform motion). If t ≥ 0, x ∈ R and E ∈ B, we let pt (x, E) = χE (x + vt), where v is a constant, and χE (y) = 1 if y ∈ E and = 0 if y ∈ E. This process, starting at x, moves deterministically with constant velocity v. Example 3.2 (Poisson process). If t ≥ 0, x ∈ R and E ∈ B, we let pt (x, E) = e−λ t

∞  (λt)n χE (x + n), n! n=0

where λ is a positive constant. This process, starting at x, advances one unit by jumps, and the probability of n jumps during the time 0 and t is equal to e−λ t (λt)n /n!. Example 3.3 (Brownian motion). If t > 0, x ∈ R and E ∈ B, we let    1 (y − x)2 pt (x, E) = √ exp − dy, 2t 2πt E and p0 (x, E) = χE (x). This is a mathematical model of one-dimensional Brownian motion. Its character is quite different from that of the Poisson process; the transition function pt (x, E) satisfies the condition pt (x, [x − ε, x + ε]) = 1 − o(t)

as t ↓ 0,

for all ε > 0 and x ∈ R. This means that the process never stands still, as does the Poisson process. Indeed, this process changes state not by jumps but by continuous motion. A Markov process with this property is called a diffusion process. Example 3.4 (Brownian motion with constant drift). If t > 0, x ∈ R and E ∈ B, we let    1 (y − mt − x)2 pt (x, E) = √ exp − dy, 2t 2πt E and p0 (x, E) = χE (x), where m is a constant.

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71

This represents Brownian motion with a constant drift of magnitude m superimposed; the process can be represented as {xt + mt}, where {xt } is Brownian motion on R. Example 3.5 (Cauchy process). If t > 0, x ∈ R and E ∈ B, we let  1 t pt (x, E) = dy, 2 π E t + (y − x)2 and p0 (x, E) = χE (x). This process can be thought of as the “trace” on the real line of trajectories of two-dimensional Brownian motion, and it moves by jumps (see [69, Lemma 2.12]). More precisely, if B1 (t) and B2 (t) are two independent Brownian motions and if T is the first passage time of B1 (t) to x, then B2 (T ) has the Cauchy density 1 |x| , −∞ < y < ∞. π x2 + y 2 Here are three more examples of diffusion processes on the half line R+ = [0, ∞) in which we must take account of the effect of the boundary point 0: Example 3.6 (Reflecting barrier Brownian motion). If t > 0, x ∈ R+ and E ∈ B, we let        1 (y − x)2 (y + x)2 pt (x, E) = √ exp − exp − dy + dy , 2t 2t 2πt E E and p0 (x, E) = χE (x). This represents Brownian motion with a reflecting barrier at x = 0; the process may be represented as {|xt |}, where {xt } is Brownian motion on R. Indeed, since {|xt |} goes from x to y if {xt } goes from x to ±y due to the symmetry of the transition function in Example 3.3 about x = 0, it follows that (see Figure 3.8 below) pt (x, E) = Px {|xt | ∈ E}        (y − x)2 (y + x)2 1 exp − exp − =√ dy + dy . 2t 2t 2πt E E Example 3.7 (Absorbing barrier Brownian motion). If t > 0, x ∈ K = [0, ∞) and E ∈ B, we let        1 (y − x)2 (y + x)2 exp − exp − pt (x, E) = √ dy − dy , 2t 2t 2πt E E and

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3 Markov Processes and Feller Semigroups y 2

1

reflection

−x

0

x

Fig. 3.8. The reflecting barrier Brownian motion

p0 (x, E) = χE (x). This represents Brownian motion with an absorbing barrier at x = 0; a Brownian particle dies at the first moment when it hits the boundary point x = 0 (see Figure 3.9 below). Namely, the boundary point 0 of K is the terminal point.

y 2 absorbtion

−x

1

0

x

Fig. 3.9. The absorbing barrier Brownian motion

Example 3.8 (Sticking barrier Brownian motion). If t > 0, x ∈ R+ and E ∈ B, we let        1 (y − x)2 (y + x)2 pt (x, E) = √ exp − exp − dy − dy 2t 2t 2πt E E  2    x z 1 exp − + 1− √ dz χE (0), 2t 2πt −x and

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73

p0 (x, E) = χE (x). This represents Brownian motion with a sticking barrier at x = 0. When a Brownian particle reaches the boundary point 0 for the first time, instead of reflecting it sticks there forever; in this case the state 0 is called a trap. 3.2.3 Path Functions of Markov Processes It is naturally interesting and important to ask the following problem: Problem. Given a Markov transition function pt (x, ·), under which conditions on pt (x, ·) does there exist a Markov process with transition function pt (x, ·) whose paths are almost surely continuous ? A Markov process X = (xt , F , Ft , Px ) is said to be right continuous provided that we have, for each x ∈ K, Px {ω ∈ Ω : the mapping t → xt (ω) is a right continuous function from [0, ∞) into K∂ } = 1. Furthermore, we say that X is continuous provided that we have, for each x ∈ K, Px {ω ∈ Ω : the mapping t → xt (ω) is a continuous function from [0, ζ(ω)) into K∂ } = 1, where ζ is the lifetime of the process X . Now we give some useful criteria for path continuity in terms of Markov transition functions (see Dynkin [33, Chapter 6], [34, p. 91, Theorem 3.5]): Theorem 3.19. Let (K, ρ) be a locally compact, separable metric space and let pt (x, ·) be a normal Markov transition function on K. (i) Assume that the following two conditions (L) and (M) are satisfied: (L) For each s > 0 and each compact E ⊂ K, we have the condition lim sup pt (x, E) = 0.

x→∂ 0≤t≤s

(M) For each ε > 0 and each compact E ⊂ K, we have the condition lim sup pt (x, K \ Uε (x)) = 0, t↓0 x∈E

where Uε (x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x. Then there exists a Markov process X with transition function pt (x, ·) whose paths are right continuous on [0, ∞) and have left-hand limits on [0, ζ) almost surely. (ii) Assume that condition (L) and the following condition (N) (replacing condition (M)) are satisfied:

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3 Markov Processes and Feller Semigroups

(N) For each ε > 0 and each compact E ⊂ K, we have the condition lim t↓0

1 sup pt (x, K \ Uε (x)) = 0, t x∈E

or equivalently sup pt (x, K \ Uε (x)) = o(t)

x∈E

as t ↓ 0.

Then there exists a Markov process X with transition function pt (x, ·) whose paths are almost surely continuous on [0, ζ). Remark 3.20. It is known (see Dynkin [33, Lemma 6.2]) that if the paths of a Markov process are right continuous, then the transition function pt (x, ·) satisfies the condition lim pt (x, Uε (x)) = 1 for all x ∈ K. t↓0

3.2.4 Stopping Times In this subsection we formulate the starting afresh property for suitable random times τ , that is, the events {ω ∈ Ω : τ (ω) < a} should depend on the process {xt } only “up to time a”, but not on the “future” after time a. This idea leads us to the following definition (see [34, Chapter III, Section 3]): Definition 3.21. Let {Ft : t ≥ 0} be an increasing family of σ-algebras in a probability space (Ω, F , P ). A mapping τ : Ω → [0, ∞] is called a stopping time or Markov time with respect to {Ft } if it satisfies the condition {τ < a} = {ω ∈ Ω : τ (ω) < a} ∈ Fa

for all a > 0.

(3.18)

for all a > 0,

(3.19)

If we introduce another condition {τ ≤ a} = {ω ∈ Ω : τ (ω) ≤ a} ∈ Fa

then condition (3.19) implies condition (3.18); hence we obtain a smaller family of stopping times. Conversely, we can prove the following lemma (see [122, Lemma 9.23]): Lemma 3.22. Assume that the family {Ft } is right-continuous, that is, , Fs for each t ≥ 0. Ft = s>t

Then condition (3.18) implies condition (3.19).

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75

Therefore, we find that conditions (3.18) and (3.19) are equivalent provided that the family {Ft } is right-continuous. If τ is a stopping time with respect to the right-continuous family {Ft } of σ-algebras, we let Fτ = {A ∈ F : A ∩ {τ ≤ a} ∈ Fa

for all a > 0} .

Intuitively, we may think of Fτ as the “past” up to the random time τ . Then we have the following lemma (see [122, Lemma 9.24]): Lemma 3.23. Fτ is a σ-algebra. Now we list some elementary properties of stopping times and their associated σ-algebras: (i) Any non-negative constant mapping is a stopping time. More precisely, if τ ≡ t0 for some constant t0 ≥ 0, then it follows that τ is a stopping time and that Fτ reduces to Ft0 . (ii) If {τn } is a finite or denumerable collection of stopping times for the family {Ft }, then it follows that τ = inf τn n

is also a stopping time. (iii) If {τn } is a finite or denumerable collection of stopping times for the family {Ft }, then it follows that τ = sup τn n

is also a stopping time. (iv) If τ is a stopping time and t0 is a positive constant, then it follows that τ + t0 is also a stopping time. (v) Let τ1 and τ2 be stopping times for the family {Ft } such that τ1 ≤ τ2 on Ω. Then it follows that Fτ 1 ⊂ Fτ 2 . This is a generalization of the monotonicity of the family {Ft }. (vi) Let {τn }∞ n=1 be a sequence of stopping times for the family {Ft } such that τn+1 ≤ τn on Ω. Then it follows that the limit τ = lim τn = inf τn n→∞

n≥1

is a stopping time and further that , Fτ n . Fτ = n≥1

This property generalizes the right-continuity of the family {Ft }.

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3 Markov Processes and Feller Semigroups

3.2.5 Definition of Strong Markov Processes A Markov process is called a strong Markov process if the “starting afresh” property holds not only for every fixed moment but also for suitable random times. In this subsection, following Dynkin [34, Chapter III, Section 3] we formulate precisely this “strong” Markov property, and give a useful criterion for the strong Markov property. Let (K, ρ) be a locally compact, separable metric space, and let K∂ = K ∪ {∂} be its one-point compactification. Namely, we add a new point ∂ to the locally compact space K as the point at infinity if K is not compact, and as an isolated point if K is compact. Let X = (xt , F , Ft , Px ) be a Markov process. For each t ∈ [0, ∞], we define a mapping Φt : [0, t] × Ω −→ K∂ by the formula Φt (s, ω) = xs (ω). A Markov process X = (xt , F , Ft , Px ) is said to be progressively measurable with respect to {Ft } if the mapping Φt is B[0,t] × Ft /B∂ -measurable for each t ∈ [0, ∞], that is, if we have the condition Φ−1 t (E) = {Φt ∈ E} ∈ B[0,t] × Ft

for all E ∈ B∂ .

Here B[0,t] is the σ-algebra of all Borel sets in the interval [0, t] and B∂ is the σ-algebra in K∂ generated by B. It should be noticed that if X is progressively measurable and if τ is a stopping time, then the mapping xτ : ω → xτ (ω) (ω) is Fτ /B∂ - measurable. Definition 3.24. We say that a progressively measurable Markov process X = (xt , F , Ft , Px ) has the strong Markov property with respect to {Ft } if the following condition is satisfied: For all h ≥ 0, x ∈ K∂ , E ∈ B∂ and all stopping times τ , we have the formula Px {xτ +h ∈ E | Fτ } = ph (xτ , E), or equivalently,  Px (A ∩ {xτ +h ∈ E}) = A

ph (xτ (ω) (ω), E) dPx (ω)

for all A ∈ Fτ .

This expresses the idea of “starting afresh” at random times (cf. Definition 3.16). We shall state a simple criterion for the strong Markov property in terms of transition functions. Let (K, ρ) be a locally compact, separable metric space. We add a point ∂ to the metric space K as the point at infinity if K is not compact, and as

3.2 Elements of Markov Processes

77

an isolated point if K is compact; so the space K∂ = K ∪ {∂} is compact (see Figure 3.5). Let C(K) be the space of real-valued, bounded continuous functions f (x) on K; the space C(K) is a Banach space with the supremum norm

f ∞ = sup |f (x)|. x∈K

We say that a function f ∈ C(K) converges to zero as x → ∂ if, for each ε > 0, there exists a compact subset E of K such that |f (x)| < ε

for all x ∈ K \ E,

and we then write limx→∂ f (x) = 0. We let   C0 (K) = f ∈ C(K) : lim f (x) = 0 . x→∂

The space C0 (K) is a closed subspace of C(K); hence it is a Banach space. Note that C0 (K) may be identified with C(K) if K is compact. We introduce a useful convention as follows: Any real-valued function f (x) on K is extended to the space K∂ = K ∪ {∂}by setting f (∂) = 0. From this point of view, the space C0 (K) is identified with the subspace of C(K∂ ) which consists of all functions f (x) satisfying the condition f (∂) = 0: C0 (K) = {f ∈ C(K∂ ) : f (∂) = 0} . Furthermore, we can extend a Markov transition function pt (x, ·) on K to a Markov transition function pt (x, ·) on K∂ by the formulas: ⎧  ⎪ for all x ∈ K and E ∈ B, ⎨pt (x, E) = pt (x, E)  pt (x, {∂}) = 1 − pt (x, K) for all x ∈ K, ⎪ ⎩   pt (∂, K) = 0, pt (∂, {∂}) = 1. Intuitively, this means that a Markovian particle moves in the space K until it “dies” at the time it reaches the point ∂; hence the point ∂ is called the terminal point. Now we introduce some conditions on the measures pt (x, ·) related to continuity in x ∈ K, for fixed t ≥ 0 (see [34, Chapter II, Section 5]): Definition 3.25. (i) A Markov transition function pt (x, ·) is called a Feller function if the function  Tt f (x) = pt (x, dy)f (y) K

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3 Markov Processes and Feller Semigroups

is a continuous function of x ∈ K whenever f is in C(K), that is, if we have the condition f ∈ C(K) =⇒ Tt f ∈ C(K). (ii) We say that pt (x, ·) is a C0 -function if the space C0 (K) is an invariant subspace of C(K) for the operators Tt : f ∈ C0 (K) =⇒ Tt f ∈ C0 (K). Remark 3.26. The Feller property is equivalent to saying that the measures pt (x, ·) depend continuously on x ∈ K in the usual weak topology, for every fixed t ≥ 0. The next result gives a useful criterion for the strong Markov property ([33, Theorem 5.10], [34, p. 99, Theorem 3.10])): Theorem 3.27. Assume that the transition function of some right-continuous Markov process has the C0 -property. Then it is a strong Markov process. 3.2.6 Strong Markov Property and Uniform Stochastic Continuity In this subsection, following Dynkin [33] and [34] we introduce the basic notion of uniform stochastic continuity of transition functions, and give simple criteria for the strong Markov property in terms of transition functions (see Figure 3.10 below). Let (K, ρ) be a locally compact, separable metric space. We begin with the following definition (see [34, p. 92, Condition M (Γ )]): Definition 3.28. A transition function pt on K is said to be uniformly stochastically continuous on K if it satisfies the following condition: For each ε > 0 and each compact E ⊂ K, we have the assertion lim sup [1 − pt (x, Uε (x))] = 0, t↓0 x∈E

(3.20)

where Uε (x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x. It should be noticed that every uniformly stochastically continuous transition function pt is normal and satisfies condition (M) in Theorem 3.19. Therefore, by combining part (i) of Theorem 3.19 and Theorem 3.27 we obtain the following theorem (see [34, p. 92, Theorem 3.7], [122, Theorem 9.28]): Theorem 3.29. If a uniformly stochastically continuous, C0 transition function satisfies condition (L), then it is the transition function of some strong Markov process whose paths are right-continuous and have no discontinuities other than jumps.

3.3 Markov Transition Functions and Feller Semigroups

uniform stochastic continuity + condition (L)

right-continuous Markov process

79

C0 -property

strong Markov process

Fig. 3.10. A functional analytic approach to strong Markov processes in Theorems 3.19, 3.27 and 3.29

Theorems 3.19, 3.27 and 3.29 can be visualized as follows. A continuous strong Markov process is called a diffusion process. The next result states a sufficient condition for the existence of a diffusion process with a prescribed transition function (see [34, p. 91, Theorem 3.5], [122, Theorem 9.29]): Theorem 3.30. If a uniformly stochastically continuous, C0 transition function satisfies conditions (L) and (N), then it is the transition function of some diffusion process. This theorem is an immediate consequence of part (ii) of Theorem 3.19 and Theorem 3.29.

3.3 Markov Transition Functions and Feller Semigroups This section provides a brief description of basic definitions and results about Markov processes and a class of semigroups (Feller semigroups) associated with Markov processes. The semigroup approach to Markov processes can be traced back to the pioneering work of Feller [39] and [40] in early 1950s (cf. [15], [98], [116]). The Feller or C0 -property deals with continuity of a Markov transition function pt (x, E) in x, and does not, by itself, have no concern with continuity in t. We give a necessary and sufficient condition on pt (x, E) in order that its associated operators {Tt }t≥0 , defined by the formula  pt (x, dy) f (y) for f ∈ C0 (K), (3.21) Tt f (x) = K

is strongly continuous in t on the space C0 (K): lim Tt+s f − Tt f ∞ = 0 s↓0

for every f ∈ C0 (K).

(3.22)

Then we have the following theorem (cf. [114, Theorem 9.2.3]; [122, Theorem 9.33]):

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3 Markov Processes and Feller Semigroups

Theorem 3.31. Let pt (x, ·) be a C0 -transition function on a locally compact, separable metric space K. Then the associated operators {Tt }t≥0 , defined by formula (3.21) is strongly continuous in t on C0 (K) if and only if pt (x, ·) is uniformly stochastically continuous on K and satisfies condition (L). Proof. (i) The “if” part: Since continuous functions with compact support are dense in C0 (K), it suffices to prove the strong continuity of {Tt } at t = 0: lim Tt f − f ∞ = 0

(3.23)

t↓0

for all such functions f . For any compact subset E of K containing the support supp f of f , we have the inequality

Tt f − f ∞ ≤ sup |Tt f (x) − f (x)| + sup |Tt f (x)| x∈E

(3.24)

x∈K\E

≤ sup |Tt f (x) − f (x)| + f ∞ · sup pt (x, supp f ). x∈E

x∈K\E

However, condition (L) implies that, for each ε > 0 we can find a compact subset E of K such that, for all sufficiently small t > 0, sup pt (x, supp f ) < ε.

(3.25)

x∈K\E

On the other hand, we have, for each δ > 0, Tt f (x) − f (x)  = pt (x, dy)(f (y) − f (x)) Uδ (x)  pt (x, dy) (f (y) − f (x)) − f (x) (1 − pt (x, K)) , + K\Uδ (x)

and hence sup |Tt f (x) − f (x)|

x∈E

≤

sup |f (y) − f (x)| + 3 f ∞ sup [1 − pt (x, Uδ (x))] . x∈E

ρ(x,y) 0, sup |Tt f (x) − f (x)| < ε(1 + 3 f ∞ ).

(3.26)

x∈E

Therefore, by carrying inequalities (3.25) and (3.26) into inequality (3.24) we obtain that, for all sufficiently small t > 0,

Tt f − f ∞ < ε(1 + 4 f ∞). This proves the desired formula (3.23), that is, the strong continuity of {Tt }. (ii) The “only if” part: For any x ∈ K and ε > 0, we define a continuous function fx (y) by the formula (see Figure 3.11 below) ⎧ ⎨1 − 1 ρ(x, y) if ρ(x, y) ≤ ε, (3.27) fx (y) = ε ⎩0 if ρ(x, y) > ε. Let E be an arbitrary compact subset of K. Then, for all sufficiently small fx

ε

x

ε

Fig. 3.11. The function fx

ε > 0, the functions fx , x ∈ E, are in C0 (K) and satisfy the condition

fx − fz ∞ ≤

1 ρ(x, z) for all x, z ∈ E. ε

(3.28)

However, for any δ > 0, by the compactness of E we can find a finite number of points x1 , x2 , . . ., xn of E such that E=

n "

Uδε/4 (xk ),

k=1

and hence min ρ(x, xk ) ≤

1≤k≤n

δε 4

for all x ∈ E.

Thus, by combining this inequality with inequality (3.28) with z := xk we obtain that δ for all x ∈ E. (3.29) min fx − fxk ∞ ≤ 1≤k≤n 4

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3 Markov Processes and Feller Semigroups

Now we have, by formula (3.27),  0 ≤ 1 − pt (x, Uε (x)) ≤ 1 − pt (x, dy)fx (y) K∂

= fx (x) − Tt fx (x) ≤ fx − Tt fx ∞ ≤ fx − fxk ∞ + fxk − Tt fxk ∞ + Tt fxk − Tt fx ∞ ≤ 2 fx − fxk ∞ + fxk − Tt fxk ∞

for all x ∈ E.

In view of inequality (3.29), the first term on the last inequality is bounded by δ/2 for the right choice of k. Furthermore, it follows from the strong continuity (3.23) of {Tt } that the second term tends to zero as t ↓ 0 for each k = 1, 2, . . ., n. Consequently, we have, for all sufficiently small t > 0, sup [1 − pt (x, Uε (x))] ≤ δ.

x∈E

This proves the desired condition (3.20), that is, the uniform stochastic continuity of pt (x, ·). Finally, it remains to verify condition (L). Our proof is based on a reduction to absurdity. Assume, to the contrary, that: For some s > 0 and some compact E ⊂ K, there exist a positive constant ε0 , a sequence {tk }, tk ↓ t (0 ≤ t ≤ s) and a sequence {xk }, xk → ∂, such that (3.30) ptk (xk , E) ≥ ε0 . Now we take a relatively compact subset U of K containing E, and let (see Figure 3.12 below) f (x) =

ρ(x, K \ U ) . ρ(x, E) + ρ(x, K \ U ) f

E U

Fig. 3.12. The function f (x)

Then it follows that the function f (x) is in C0 (K) and satisfies the condition

3.3 Markov Transition Functions and Feller Semigroups

83

 pt (x, dy) f (y) ≥ pt (x, E) ≥ 0.

(Tt f ) (x) = K

Therefore, by combining this inequality with inequality (3.30) we obtain that (Ttk f ) (xk ) ≥ ptk (xk , E) ≥ ε0

for all k ∈ N.

(3.31)

However, we have the inequality (Ttk f ) (xk ) ≤ Ttk f − Tt f ∞ + (Tt f ) (xk ) for all k ∈ N.

(3.32)

Since the semigroup {Tt } is strongly continuous and since we have the assertion  lim (Tt f ) (xk ) = pt (∂, dy) f (y) = f (∂) = 0, k→∞

K∂

we can let k → ∞ in inequality (3.32) to obtain that lim sup (Ttk f ) (xk ) = 0. k→∞

This contradicts inequality (3.31). The proof of Theorem 3.31 is now complete.   Now we are in a position to define the following definition: Definition 3.32. A family {Tt }t≥0 of bounded linear operators acting on the Banach space C0 (K) is called a Feller semigroup on the state space K if it satisfies the following three conditions (i), (ii) and (iii): (i) Tt+s = Tt · Ts , t, s ≥ 0 (the semigroup property); T0 = I. (ii) The family {Tt } is strongly continuous in t for all t ≥ 0: lim Tt+s f − Tt f ∞ = 0 s↓0

for each f ∈ C0 (K).

(iii) The family {Tt } is non-negative and contractive on C0 (K): f ∈ C0 (K), 0 ≤ f (x) ≤ 1

on K =⇒ 0 ≤ Tt f (x) ≤ 1

on K.

Rephrased, Theorem 3.31 gives a characterization of Feller semigroups in terms of Markov transition functions (see Figure 3.13 below): Theorem 3.33 (Dynkin). If pt (x, ·) is a uniformly stochastically continuous, C0 -transition function on a locally compact, separable metric space K and satisfies condition (L), then its associated operators {Tt }t≥0 , defined by formula (3.21), form a Feller semigroup on the state space K. Conversely, if {Tt }t≥0 is a Feller semigroup on the state space K, there exists a uniformly stochastically continuous, C0 -transition function pt (x, ·) on K, satisfying condition (L), such that formula (3.21) holds true. The most important applications of Theorem 3.33 are of course in the second statement.

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3 Markov Processes and Feller Semigroups

pt (x, ·) : uniform stochastic continuity + C0 -property + condition (L)

{Tt } : Feller semigroup on C0 (K) Fig. 3.13. A functional analytic approach to strong Markov processes in Theorems 3.31 and 3.33

3.4 Generation Theorems for Feller Semigroups In this section we prove various generation theorems of Feller semigroups by using the Hille–Yosida theory of semigroups (Theorems 3.34 and 3.36) which form a functional analytic background for the proof of Theorem 1.6. If {Tt }t≥0 is a Feller semigroup on the state space K, we define its infinitesimal generator A by the formula Au = lim t↓0

Tt u − u t

for u ∈ C0 (K),

(3.33)

provided that the limit (3.33) exists in the space C0 (K). More precisely, the generator A is a linear operator from C0 (K) into itself defined as follows: (1) The domain D(A) of A is the set D(A) = {u ∈ C0 (K) : the limit (3.33) exists} . Tt u − u for every u ∈ D(A). t The next theorem is a version of the Hille–Yosida theorem adapted to the present context (cf. [114, Theorem 9.3.1 and Corollary 9.3.2]; [122, Theorem 9.35]):

(2) Au = limt↓0

Theorem 3.34 (Hille–Yosida). (i) Let {Tt }t≥0 be a Feller semigroup on the state space K and let A be its infinitesimal generator. Then we have the following four assertions (a) through (d): (a) The domain D(A) is dense in the space C0 (K). (b) For each α > 0, the equation (αI − A)u = f has a unique solution u in D(A) for any f ∈ C0 (K). Hence, for each α > 0 the Green operator (αI − A)−1 : C0 (K) → C0 (K) can be defined by the formula u = (αI − A)−1 f

for f ∈ C0 (K).

3.4 Generation Theorems for Feller Semigroups

85

(c) For each α > 0, the operator (αI − A)−1 is non-negative on C0 (K): f ∈ C0 (K), f (x) ≥ 0

on K =⇒ (αI − A)−1 f (x) ≥ 0

on K.

(d) For each α > 0, the operator (αI − A)−1 is bounded on C0 (K) with norm

(αI − A)−1 ≤

1 . α

(ii) Conversely, if A is a linear operator from C0 (K) into itself satisfying condition (a) and if there is a non-negative constant α0 such that, for all α > α0 , conditions (b), (c) and (d) are satisfied, then A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on the state space K. Proof. In view of the Hille–Yosida theory (see [147, Chapter IX, Section 7]), it suffices to show that the semigroup {Tt }t≥0 is non-negative if and only if its resolvents (Green operators) {(αI − A)−1 }α>α0 are non-negative. The “only if” part is an immediate consequence of the following expression of the resolvent (αI − A)−1 in terms of the semigroup {Tt }:  ∞ exp[−α t] Tt dt for α > 0. (αI − A)−1 = 0

On the other hand, the “if” part follows from the expression of the semigroup Tt (α) in terms of the Yosida approximation Jα = α(αI − A)−1 : Tt (α) = exp[−α t] exp [α t Jα ] = exp[−α t]

∞  (αt)n n Jα , n! n=0

and the definition of the semigroup Tt : Tt = lim Tt (α). α→∞

The proof of Theorem 3.34 is complete.   Corollary 3.35. Let K be a compact metric space and let A be the infinitesimal generator of a Feller semigroup on the state space K. Assume that the constant function 1 belongs to the domain D(A) of A and that we have, for some constant c, (A1)(x) ≤ −c on K. (3.34) Then the operator A = A + c I is the infinitesimal generator of some Feller semigroup on the state space K. Proof. It follows from an application of part (i) of Theorem 3.34 that the operators −1 −1 (αI − A ) = ((α − c) I − A)

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3 Markov Processes and Feller Semigroups

are defined and non-negative on the whole space C(K), for all α > c. However, in view of inequality (3.34) we obtain that α ≤ α − (A1 + c) = (αI − A ) 1

on K,

so that α(αI − A )−1 1 ≤ (αI − A )−1 ((αI − A )1) = 1 on K. Hence we have, for all α > c,

(αI − A )−1 = (αI − A )−1 1 ∞ ≤

1 . α

Therefore, by applying part (ii) of Theorem 3.34 to the operator A we find that A is the infinitesimal generator of some Feller semigroup on the state space K. The proof of Corollary 3.35 is complete.   The Hille–Yosida theory of semigroups via the Laplace transform can be visualized as in Figure 3.14 below.

Tt = etA via the Laplace transform (αI − A)−1 =

∞ −αt tA e e 0

dt =

∞ −αt e 0

Tt dt

Fig. 3.14. The Hille–Yosida theory of semigroups via the Laplace transform

Now we write down explicitly the infinitesimal generators of Feller semigroups associated with the transition functions in Examples 3.1 through 3.8 (cf. [35]). Example 3.9 (Uniform motion). K = R and  D(A) = {f ∈ C0 (K) : f  ∈ C0 (K)}, Af = vf  for every f ∈ D(A). Example 3.10 (Poisson process). K = R and  D(A) = C0 (K), Af (x) = λ(f (x + 1) − f (x)) for every f ∈ D(A). The operator A is not “local”; the value Af (x) depends on the values f (x) and f (x + 1). This reflects the fact that the Poisson process changes state by jumps.

3.4 Generation Theorems for Feller Semigroups

87

Example 3.11 (Brownian motion). K = R and ⎧ ⎨D(A) = {f ∈ C0 (K) : f  ∈ C0 (K), f  ∈ C0 (K)}, 1 ⎩Af = f  for every f ∈ D(A). 2 The operator A is “local”, that is, the value Af (x) is determined by the values of f in an arbitrary small neighborhood of x. This reflects the fact that Brownian motion changes state by continuous motion. Example 3.12 (Brownian motion with constant drift). K = R and ⎧ ⎨D(A) = {f ∈ C0 (K) : f  ∈ C0 (K), f  ∈ C0 (K)} , 1 ⎩Af = f  + mf  for every f ∈ D(A). 2 Example 3.13 (Cauchy process). K = R and, the domain D(A) contains C 2 functions on K with compact support, and the infinitesimal generator A is of the form  dy 1 +∞ (f (x + y) − f (x)) 2 . Af (x) = π −∞ y The operator A is not “local”, which reflects the fact that the Cauchy process changes state by jumps. More precisely, see Example 4.8 and Table 4.4 with n := 1 and α := 1 in Chapter 4. Example 3.14 (Reflecting barrier Brownian motion). K = [0, ∞) and ⎧ ⎨D(A) = {f ∈ C0 (K) : f  ∈ C0 (K), f  ∈ C0 (K), f  (0) = 0} , 1 ⎩Af = f  for every f ∈ D(A). 2 Example 3.15 (Sticking barrier Brownian motion). K = [0, ∞) and ⎧ ⎨D(A) = {f ∈ C0 (K) : f  ∈ C0 (K), f  ∈ C0 (K), f  (0) = 0} , 1 ⎩Af = f  for every f ∈ D(A). 2 Finally, here are two more examples where it is difficult to begin with a transition function and the infinitesimal generator is the basic tool of describing the process. Example 3.16 (Sticky barrier Brownian motion). K = [0, ∞) and ⎧ ⎨D(A) = {f ∈ C0 (K) : f  ∈ C0 (K), f  ∈ C0 (K), f  (0) − αf  (0) = 0} , 1 ⎩Af = f  for every f ∈ D(A). 2

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3 Markov Processes and Feller Semigroups

Here α is a positive constant. This process may be thought of as a “combination” of the reflecting and sticking Brownian motions. The reflecting and sticking cases are obtained by letting α → 0 and α → ∞, respectively. Example 3.17 (Absorbing barrier Brownian motion). K = [0, ∞) where the boundary point 0 is identified with the point at infinity ∂. ⎧ ⎨D(A) = {f ∈ C0 (K) : f  ∈ C0 (K), f  ∈ C0 (K), f (0) = 0} , 1 ⎩Af = f  for every f ∈ D(A). 2 This represents Brownian motion with an absorbing barrier at x = 0; a Brownian particle “dies” at the first moment when it hits the boundary x = 0. Namely, the point 0 is the terminal point. It is worth pointing out here that a strong Markov process cannot stay at a single position for a positive length of time and then leave that position by continuous motion; it must either jump away or leave instantaneously. We give a simple example of a strong Markov process which changes state not by continuous motion but by jumps when the motion reaches the boundary. Example 3.18. K = [0, ∞). ⎧ ⎪ D(A) = {f ∈ C0 (K) ∩ C 2 (K) : f  ∈ C0 (K), f  ∈ C0 (K), ⎪ ⎨ +∞ f  (0) = 2c 0 (f (y) − f (0))dF (y)}, ⎪ ⎪ ⎩Af = 1 f  for every f ∈ D(A). 2 Here c is a positive constant and F is a distribution function on the interval (0, ∞). This process may be interpreted as follows. When a Brownian particle reaches the boundary x = 0, it stays there for a positive length of time and then jumps back to a random point, chosen with the function F , in the interior (0, ∞). The constant c is the parameter in the “waiting time” distribution at the boundary x = 0. We remark that the boundary condition  ∞  (f (y) − f (0)) dF (y) f (0) = 2c 0

depends on the values of f far away from the boundary x = 0, unlike the boundary conditions in Examples 3.14 through 3.17. Although Theorem 3.34 asserts precisely when a linear operator A is the infinitesimal generator of some Feller semigroup, it is usually difficult to verify conditions (b) through (d). So we give useful criteria in terms of the maximum principle (see [15], [98], [92], [114, Theorem 9.3.3 and Corollary 9.3.4]; [122, Theorem 9.50]):

3.4 Generation Theorems for Feller Semigroups

89

Theorem 3.36 (Hille–Yosida–Ray). If K is a compact metric space, then we have the following two assertions (i) and (ii): (i) Let B be a linear operator from C(K) = C0 (K) into itself, and assume that: (α) The domain D(B) of B is dense in the space C(K). (β) There exists an open and dense subset K0 of K such that if a function u ∈ D(B) takes a positive maximum at a point x0 of K0 , then we have the inequality Bu(x0 ) ≤ 0. Then the operator B is closable in the space C(K). (ii) Let B be as in part (i), and further assume that: (β  ) If a function u ∈ D(B) takes a positive maximum at a point x of K, then we have the inequality Bu(x ) ≤ 0. (γ) For some α0 ≥ 0, the range R (α0 I − B) of α0 I − B is dense in the space C(K). Then the minimal closed extension B of B is the infinitesimal generator of some Feller semigroup on the state space K. Proof. (i) It suffices to show that: {un } ⊂ D(B), un → 0 and Bun → v

in C(K) =⇒ v = 0.

Our proof is based on a reduction to absurdity. By replacing v by −v if necessary, we assume, to the contrary, that: The function v(x) takes a positive value at some point of K. Then, since K0 is open and dense in K, we can find a point x0 of K0 , a neighborhood U of x0 contained in K0 and a positive constant ε such that we have, for all sufficiently large n, Bun (x) > ε

for all x ∈ U .

(3.35)

On the other hand, by condition (α) there exists a function h ∈ D(B) such that  h(x0 ) > 1, h(x) < 0 for all x ∈ K \ U . Therefore, since un → 0 in C(K), it follows that the function un (x) = un (x) + satisfies the conditions

εh(x) 1 + Bh ∞

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3 Markov Processes and Feller Semigroups

εh(x0 ) > 0, 1 + Bh ∞ εh(x) < 0 for all x ∈ K \ U , un (x) = un (x) + 1 + Bh ∞

un (x0 ) = un (x0 ) +

if n is sufficiently large. This implies that the function un ∈ D(B) takes its positive maximum at a point xn of U ⊂ K0 . Hence we have, by condition (β), (Bun ) (xn ) ≤ 0. However, it follows from inequality (3.35) that (Bun ) (xn ) = (Bun ) (xn ) + ε

(Bh)(xn ) > (Bun ) (xn ) − ε > 0. 1 + Bh ∞

This is a contradiction. (ii) We apply part (ii) of Theorem 3.34 to the minimal closed extension B of B. The proof is divided into six steps. Step 1: First, we show that u ∈ D(B), (α0 I − B) u ≥ 0 on K =⇒ u ≥ 0 on K.

(3.36)

By condition (γ), we can find a function v ∈ D(B) such that (α0 I − B) v ≥ 1 on K. Then we have, for any ε > 0,  u + εv ∈ D(B), (α0 I − B) (u + εv) ≥ ε

(3.37)

on K.

In view of condition (β  ), this implies that the function −(u(x) + εv(x)) does not take any positive maximum on K, so that u(x) + εv(x) ≥ 0

on K.

Thus, by letting ε ↓ 0 in this inequality we obtain that u(x) ≥ 0 on K. This proves the desired assertion (3.36). Step 2: It follows from assertion (3.36) that the inverse (α0 I − B)−1 of α0 I − B is defined and non-negative on the range R (α0 I − B). Moreover, it is bounded with norm    −1  (3.38) (α0 I − B)  ≤ v ∞ . Here v(x) is the function that satisfies condition (3.37).

3.4 Generation Theorems for Feller Semigroups

91

Indeed, since g = (α0 I − B)v ≥ 1 on K, it follows that, for all f ∈ C(K), − f ∞ g ≤ f ≤ f ∞g

on K.

Hence, by the non-negativity of (α0 I − B)−1 we have, for all f ∈ R (α0 I − B), − f ∞ v ≤ (α0 I − B)−1 f ≤ f ∞ v

on K.

This proves the desired inequality (3.38). Step 3: Next we show that   R α0 I − B = C(K).

(3.39)

Let f (x) be an arbitrary element of C(K). By condition (γ), we can find a sequence {un } in D(B) such that fn = (α0 I − B)un → f in C(K). Since the inverse (α0 I − B)−1 is bounded, it follows that un = (α0 I − B)−1 fn converges to some function u ∈ C(K), and hence Bun = α0 un − fn converges to α0 u − f in C(K). Thus we have, by the closedness of B,    u∈D B , Bu = α0 u − f, so that (α0 I − B)u = f. This proves the desired assertion (3.39). Step 4: Furthermore, we show that   u ∈ D B , (α0 I − B)u ≥ 0 on K =⇒ u ≥ 0 on K.

(3.40)

  Since R α0 I − B = C(K), in view of the proof of assertion (3.40) it suffices to show the following   assertion: If a function u ∈ D B takes a positive maximum at a point x of K, then we have the inequality    Bu (x ) ≤ 0. (3.41) Our proof of assertion (3.41) is based on a reduction to absurdity. Assume, to the contrary, that    Bu (x ) > 0. Since there exists a sequence {un } in D(B) such that un → u and Bun → Bu in C(K), we can find a neighborhood U of x and a positive constant ε such that, for all sufficiently large n, (Bun )(x) > ε

for all x ∈ U .

(3.42)

Furthermore, by condition (α) we can find a function h ∈ D(B) such that

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3 Markov Processes and Feller Semigroups



h(x ) > 1, h(x) < 0 for all x ∈ K \ U .

Then it follows that the function un (x) = un (x) +

εh(x) 1 + Bh ∞

satisfies the condition  un (x ) > u(x ) > 0, un (x) < u(x ) for all x ∈ K \ U , if n is sufficiently large. This implies that the function un ∈ D(B) takes its positive maximum at a point xn of U . Hence we have, by condition (β  ), (Bun ) (xn ) ≤ 0

for xn ∈ U .

However, it follows from inequality (3.42) that (Bun ) (xn ) = (Bun )(xn ) + ε

(Bh)(xn ) > (Bun )(xn ) − ε > 0. 1 + Bh ∞

This is a contradiction. Step 5: In view of Steps 3 and 4, we obtain that the inverse (α0 I − B)−1 of α0 I − B is defined on the whole space C(K), and is bounded with norm   −1  −1      1 .  α0 I − B  =  α0 I − B ∞

Step 6: Finally, we show that: For all α > α0 , the inverse (αI − B)−1 of αI − B is defined on the whole space C(K), and is non-negative and bounded with norm  −1  1   (3.43) ≤ .  αI − B α We let

Gα0 = (α0 I − B)−1 .

First, we choose a constant α1 > α0 such that (α1 − α0 ) Gα0 < 1, and let α0 < α ≤ α1 . Then the C. Neumann series (see [147, Chapter II, Section 1, Theorem 2]) $ % ∞  n n u= I+ (α0 − α) Gα0 Gα0 f n=1

3.4 Generation Theorems for Feller Semigroups

93

converges in C(K), and is a solution of the equation u − (α0 − α) Gα0 u = Gα0 f Hence we have the assertions 

for any f ∈ C(K).

  u∈D B ,   αI − B u = f.

This proves that   R αI − B = C(K),

α0 < α ≤ α1 .

(3.44)

Thus, by arguing just as in the proof of Step 1 we obtain that, for any α0 < α ≤ α1 ,     u ∈ D B , αI − B u ≥ 0 on K =⇒ u ≥ 0 on K. (3.45) By combining assertions (3.44) and (3.45), we find that, for any α0 < α ≤ α1 , the inverse (αI − B)−1 is defined and non-negative on the whole space C(K). We let  −1 for α0 < α ≤ α1 . Gα = αI − B Then it follows that the operator Gα is bounded with norm

Gα ≤

1 . α

(3.46)

  Indeed, in view of assertion (3.41) it follows that if a function u ∈ D B takes a positive maximum at a point x of K, then we have the inequality    Bu (x ) ≤ 0, so that   1 1  (αI − B)u(x ) ≤  αI − B u∞ . (3.47) x∈K α α   Similarly, if the function u ∈ D B takes a negative minimum at a point of K, then (replacing u(x) by −u(x)), we have the inequality max u(x) = u(x ) ≤

− min u(x) = max (−u(x)) ≤ x∈K

x∈K

   1  αI − B u . ∞ α

(3.48)

The desired inequality (3.46) follows from inequalities (3.47) and (3.48). Summing up, we have proved assertion (3.43) for all α0 < α ≤ α1 . Now we assume that assertion (3.43) is proved for all α0 < α ≤ αn−1 , n = 2, 3, . . .. Then, by taking αn = 2αn−1 −

α1 , 2

n ≥ 2,

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3 Markov Processes and Feller Semigroups

or equivalently αn =

  1 2n−2 + α1 2

for n ≥ 2,

we have, for all αn−1 < α ≤ αn , α − αn−1 αn − αn−1 ≤ αn−1 αn−1 1 = 1 + 22−n < 1.

(α − αn−1 ) Gαn−1 ≤

Hence assertion (3.43) for αn−1 < α ≤ αn is proved just as in the proof of assertion (3.43) for α0 < α ≤ α1 . This proves the desired assertion (3.43) for all α > α0 . Consequently, by applying part (ii) of Theorem 3.34 to the operator B we obtain that B is the infinitesimal generator of some Feller semigroup on the state space K. The proof of Theorem 3.36 is now complete.   The next corollary gives a sufficient condition in order that the infinitesimal generators of Feller semigroups are stable under bounded perturbations: Corollary 3.37. Let A be the infinitesimal generator of a Feller semigroup on a compact metric space K and let M be a bounded linear operator on the Banach space C(K) into itself. Assume that either M or A = A + M satisfies condition (β  ). Then the operator A is the infinitesimal generator of some Feller semigroup on the state space K. Proof. We apply part (ii) of Theorem 3.36 to the operator A . First, we remark that A = A + M is a densely defined, closed linear operator from C(K) into itself. Since the semigroup {Tt }t≥0 is non-negative and contractive on C(K), it follows that if a function u ∈ D(A) takes a positive maximum at a point x of K, then we have the inequality Au(x ) = lim t↓0

Tt u(x ) − u(x ) ≤ 0. t

This implies that if M satisfies condition (β  ), so does A = A + M . We let −1 Gα0 = (α0 I − A) , α0 > 0. If α0 is so large that

Gα0 M ≤ Gα0 · M ≤

M < 1, α0

then the Neumann series (see [147, Chapter II, Section 1, Theorem 2])

3.5 Reflecting Diffusion

$ u=

I+

∞ 

95

% n

(Gα0 M )

Gα0 f

n=1

converges in C(K), and is a solution of the equation u − Gα0 M u = Gα0 f

for any f ∈ C(K).

Hence we have the assertions  u ∈ D(A) = D (A ) , (α0 I − A )u = f. This proves that

R(α0 I − A ) = C(K).

Therefore, by applying part (ii) of Theorem 3.36 to the operator A we obtain that A is the infinitesimal generator of some Feller semigroup on the state space K. The proof of Corollary 3.37 is complete.  

3.5 Reflecting Diffusion Let D be a bounded domain in Euclidean space RN with smooth boundary ∂D, and a fixed abstract point ∂ is adjoined to the closure D as an isolated point. We consider the following homogeneous Neumann problem for the Laplacian Δ: ⎧ ⎨Δu = f in D, ∂u ⎩ = 0 on ∂D. ∂n Sato and Ueno [98] constructed the following Markov process X = (xt , W, Bt , B, Px, x ∈ D ∪ {∂}) on the state space D ∪ {∂} (see [98, Theorem 7.1]): (1) Let W be the space of right-continuous with left limits functions w : [0, +∞] −→ D ∪ {∂} with coordinates xt (w) = w(t). (2) Let Bt = σ (xs : 0 ≤ s ≤ t) be the smallest σ-field of subsets of W with respect to which all mappings {xs : 0 ≤ s ≤ t} are measurable and let B = σ (xs : 0 ≤ s < ∞) be the smallest σ-field of subsets of W with respect to which all mappings {xs : 0 ≤ s < ∞} are measurable, respectively.

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3 Markov Processes and Feller Semigroups

(3) A random variable ζ : W → [0, +∞], called the lifetime, such that  ∈ D for 0 ≤ t < ζ(w), xt (w) = ∂ for ζ(w) ≤ t ≤ +∞. The sample path xt (w) is continuous for 0 ≤ t < ζ(w). (4) For each t ∈ [0, +∞], a pathwise shift mapping θt : W → W defined by the formula θt w(s) = w(t+ s) for all w ∈ W  . (5) The system of measures Px , x ∈ D ∪ {∂} on (W, B) such that: (a) Px (B) is B(D ∪ {∂})-measurable in x, for each B ∈ B. (b) Px ({w  ∈ W : x0(w) = x}) = 1 for each x ∈ D. (c) Px θt−1 (B) | Bt = Pxt (B) for each B ∈ B. More precisely, the formula Ex (f ◦ θt | Bt ) = Ext (f ) holds true for every B-measurable, bounded function f on D, with Px probability one. (d) The system Px ({w ∈ W : xt (w) = y}) = p(t, x, dy) is called the system of transition probabilities. (6) Px ({w ∈ W : ζ(w) = +∞}) = 1 for each x ∈ D. Namely, this process is conservative. (7) A transition semigroup of linear operators  Tt f (x) := Ex (f (xt )) = p(t, x, dy) f (y) D

forms a Feller semigroup on the state space D, and its infinitesimal generator A is equal to the minimal closed extension ΔN in the Banach space C(D) of the operator ΔN defined as follows: (a) The domain D (ΔN ) of ΔN is the space   ∂u 2 D (ΔN ) = u ∈ C (D) : = 0 on ∂D . (3.49) ∂n (b) ΔN u = Δu for every u ∈ D (ΔN ). The Markov process   X = xt , W, Bt , B, Px , x ∈ D ∪ {∂} is called the reflecting diffusion on D.

3.6 Local Time on the Boundary for the Reflecting Diffusion

97

3.6 Local Time on the Boundary for the Reflecting Diffusion Following P. L´evy [76], Sato and Ueno [98] constructed the local time τ (t) on the boundary for the reflecting diffusion X = (xt , W, Bt , B, Px , x ∈ D ∪ {∂}) (see [98, Theorem 7.2]): Roughly speaking, the local time τ (t) can be defined by the formula τ (t, w) = lim ρ↓0

1 ρ

 0

t

χDρ (xs (w)) ds

for every w ∈ W ,

where   Dρ = x ∈ D : dist (x, ∂D) < ρ  1 if x ∈ Dρ , χDρ (x) = 0 otherwise.

for ρ > 0,

The local time τ (t) enjoys the following four properties (1) through (4): (1) Px ({w ∈ W : limt→+∞ τ (t, w) = +∞}) = 1 if x ∈ D. (2) Px ({w ∈ W : τ (t, w) > 0 for all t > 0}) = 1 if and only if x ∈ ∂D. (3) τ (t, w) is a continuous, non-negative additive functional of X . More precisely, we have the assertions: (3a) τ (t, w) is measurable on the sample space {xs : 0 ≤ s ≤ t}, for each t ≥ 0. (3b) 0 = τ (0, ω) ≤ τ (t, w) and τ (t, w) is continuous in t. (3c) For all x ∈ D, we have the formula  Px {ω ∈ W : τ (t + s, w) = τ (t, w) + τ (s, θt (w))  for each t, s ≥ 0} = 1. (4) τ (t, w) increases at t only when xt (w) is on the boundary ∂D. Intuitively, τ (t, w) is the sojourn time (or occupation time) of a sample path xs (w) on the boundary ∂D up to time t. Let τ −1 (t) is the right-continuous inverse of the local time τ (t) on the boundary for the reflecting diffusion   X = xt , W, Bt , B, Px , x ∈ D ∪ {∂} , where

τ −1 (t, w) = inf {s ≥ 0 : τ (s, w) ≥ t} −1

for every w ∈ W .

The inverse local time τ (t, w) is the amount of real time spent by a Markovian particle in the space D necessary to realize that its sojourn time on the boundary ∂D exceeds t. Table 3.1 below gives a bird’s-eye view of the standard time and the local time via Figure 3.15 below.

98

3 Markov Processes and Feller Semigroups

x∗τ (t,w) (w∗ ) = xt (w) τ (t, w) = 0

Fig. 3.15. The local time τ (t, w)

State space

Trajectory

Watch

Interior D

xt (w)

t (standard time)

Boundary ∂D

x∗τ (t,w) (w∗ ) = xt (w)

τ (t, w) (local time)

Table 3.1. A world watch due to P. L´evy

3.7 Notes and Comments The material of this chapter is adapted from Blumenthal–Getoor [14], Dynkin [33], [34], Dynkin–Yushkevich [35], Feller [39], [40], Ikeda–Watanabe [62], Itˆ o– McKean, Jr. [65], Lamperti [74], Revuz–Yor [94] and also Taira [114, Chapter 9] and [122, Chapters 2 and 9]. In particular, our treatment of temporally homogeneous Markov processes follows the expositions of Dynkin [33], [34] and Blumenthal–Getoor [14]. However, unlike many other books on Markov processes, this chapter focuses on the relationship among three subjects: Feller semigroups, transition functions and Markov processes. Our semigroup approach to the problem of construction of Markov processes with Ventcel’ boundary conditions is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in functional analysis methods.

3.7 Notes and Comments

99

Hille–Phillips [54] and Yosida [147] are the classics for semigroup theory of linear operators. The reader might be referred to Engel–Nagel [37], Goldstein [52] and Pazy [87] for the modern theory of semigroups of linear operators. Section 3.1: This section is adapted from Folland [42, Chapter 7] and Rudin [95, Chapter 2]. Proposition 3.4 is taken from Folland [42, Proposition 4.35] and the proof of Theorem 3.6 is adapted from Folland [42, Theorem 7.2]. A locally compact version of the Riesz–Markov representation theorem (Theorem 3.7) is taken from Folland [42, Theorem 7.17] and a compact version of the Riesz–Markov representation theorem (Theorem 3.10) is taken from Folland [42, Corollary 7.18], respectively. See also Kolmogorov–Fomin [71], Friedman [44] and Schechter [100]. Theorem 3.15 is called Prokhorov’s theorem in probability ([38, p. 104, Theorem 2.2], [94, p. 10, Theorem (5.4)]). Section 3.2: The results here are adapted from Blumenthal–Getoor [14], Dynkin [34], Lamperti [74] and Revuz–Yor [94]. Theorem 3.18 is taken from Dynkin [33, Chapter 4, Section 2], while Theorem 3.19 is taken from Dynkin [33, Chapter 6] and [34, Chapter 3, Section 2]. Theorem 3.27 is due to Dynkin [33, Theorem 5.10] and Theorem 3.29 is due to Dynkin [33, Theorem 6.3], respectively. Theorem 3.29 is a non-compact version of Lamperti [74, Chapter 8, Section 3, Theorem 1]. Subsection 3.2.6 is adapted from Lamperti [74, Chapter 9, Section 2]. Section 3.3: The semigroup approach to Markov processes can be traced back to the work of Kolmogorov [70]. It was substantially developed in the early 1950s, with Feller [39] and [40] doing the pioneering work. Our presentation here follows the book of Dynkin [34] and also part of Lamperti’s [74]. Theorem 3.31 is a non-compact version of Lamperti [74, Chapter 7, Section 7, Theorem 1]. Altomare et al. prove that a class of Feller semigroups can be approximated by iterates of modified Bernstein–Schnabl operators ([7]), which allows to infer the preservation of such properties as the Lipschitz continuity and convexity of the semigroups. Section 3.4: Theorem 3.36 is due to Sato–Ueno [98, Theorem 1.2] and Bony–Courr`ege–Priouret [15, Th´eor`eme de Hille–Yosida–Ray] (cf. [64], [92], [122]). Sections 3.5 and 3.6: These two sections are based on the talk entitled Probabilistic approach to pseudo-differential operators delivered at Special Seminar, Leibniz Universit¨at Hannover, Germany, on October 16, 2019. The material is adapted from Sato [96], Sato–Tanaka [97] and Sato–Ueno [98]. This chapter is an expanded and revised version of Chapter 2 of the second edition [121].

Part II

Pseudo-Differential Operators and Elliptic Boundary Value Problems

4 Lp Theory of Pseudo-Differential Operators

In this chapter we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators, which may be considered as a modern version of the classical potential approach. In Section 4.1 we define H¨older spaces C k+θ (Ω) and various Sobolev spaces s,p W (Ω) and H s,p (Ω) and also Besov spaces B s,p (∂Ω) on the boundary ∂Ω of a smooth domain Ω of Euclidean space Rn . It is the imbedding characteristics of Lp Sobolev spaces that render these spaces so useful in the study of partial differential equations. In the proof of Theorem 1.5 we shall make use of some imbedding properties of Lp Sobolev spaces (Theorem 4.8). Moreover, we shall need the Rellich–Kondrachov compactness theorem for function spaces of Lp type (Theorem 4.10) in the proof of Theorem 1.4. The Rellich–Kondrachov theorem is a Sobolev space version of the Bolzano–Weierstrass theorem and the Ascoli–Arzel`a theorem in calculus. In Section 4.2 we formulate Seeley’s extension theorem (Theorem 4.11), due to Seeley [103], which asserts that the functions in C ∞ (Ω) are the restrictions to Ω of functions in C ∞ (Rn ). It should be emphasized that Besov spaces B s,p (∂Ω) enter naturally in connection with boundary value problems in the framework of Sobolev spaces of Lp type. Indeed, we need to make sense of the restriction u|∂Ω to the boundary ∂Ω as an element of a Besov space on ∂Ω when u belongs to a Sobolev space on the domain Ω. In Section 4.3, we formulate the trace theorem (Theorem 4.18) and the sectional trace theorem (Theorem 4.19) that play an important role in the study of elliptic boundary value problems. In Section 4.4, we present a brief description of basic concepts and results of the theory of Fourier integral operators and pseudo-differential operators. The Lp theory of pseudo-differential operators may be considered as a modern version of the classical potential theory. Especially, pseudo-differential operators provide a constructive tool to deal with existence and smoothness of solutions of partial differential equations. The theory of pseudo-differential operators continues to be one of the most influential works in modern history © Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 4

104

4 Lp Theory of Pseudo-Differential Operators

of analysis, and is a very refined mathematical tool whose full power is yet to be exploited. In Section 4.5 we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators. We formulate the Besov space boundedness theorem due to Bourdaud [16] (Theorem 4.47) in Subsection 4.5.2 and we give a useful criterion for hypoellipticity due to H¨ormander [59] (Theorem 4.49) in Subsection 4.5.4, which will play an essential role in the proof of our main results. In Subsection 4.5.5, following Coifman–Meyer [30, Chapitre IV, Proposition 1]) we state that the distribution kernel s(x, y) n of a pseudo-differential operator S ∈ Lm 1,0 (R ) satisfies the estimate (Theorem 4.51) C |s(x, y)| ≤ for all x, y ∈ Rn and x = y. |x − y|m+n In Section 4.6, by using the Riesz–Schauder theory we prove some of the most important results about elliptic pseudo-differential operators on a manifold and their indices in the framework of Sobolev spaces (Theorems 4.53 through 4.67). These results play an important role in the study of elliptic boundary value problems in Chapter 5. The heat kernel has many important and interesting applications in partial differential equations. In Section 4.7, by calculating various convolution kernels for the Laplacian via the Laplace transform we derive Newtonian, Riesz and Bessel potentials and also the Poisson kernel for the Dirichlet boundary value problem (Theorems 4.69, 4.70 and 4.73).

4.1 Function Spaces Let Ω be a bounded domain of Euclidean space Rn with smooth boundary ∂Ω. Its closure Ω = Ω ∪ ∂Ω is an n-dimensional, compact smooth manifold with boundary. We may assume the following three conditions (a), (b) and (c) (see Theorems 4.15 and 4.16 in Section 4.3): (a) The domain Ω is a relatively compact, open subset of an n-dimensional, compact smooth manifold M without boundary in which Ω has a smooth boundary ∂Ω (see Figures 4.1 below). (b) In a neighborhood W of ∂Ω in M a normal coordinate t is chosen so that the points of W are represented as (x , t), x ∈ ∂Ω, −1 < t < 1; t > 0 in Ω, t < 0 in M \ Ω and t = 0 only on ∂Ω (see Figures 4.2 below). (c) The manifold M is equipped with a strictly positive density μ which, on W , is the product of a strictly positive density ω on ∂Ω and the Lebesgue measure dt on (−1, 1). This manifold M is called the double of Ω (see [83]). The function spaces we shall treat are the following (cf. [2], [13], [21], [43], [112], [135]):

4.1 Function Spaces

M =Ω

105

Ω = {t > 0} ∂Ω = {t = 0}

Fig. 4.1. The double M of Ω

∂Ω = {t = 0}

W {0 < t < 1} {−1 < t < 0}

Fig. 4.2. The tubular neighborhood W of the boundary ∂Ω

(i) The generalized Sobolev spaces H s,p (Ω) and H s,p (M ), consisting of all potentials of order s of Lp functions. When s is integral, these spaces coincide with the usual Sobolev spaces W s,p (Ω) and W s,p (M ), respectively. (ii) The Besov spaces B s,p (∂Ω). These are functions spaces defined in terms of the Lp modulus of continuity, and enter naturally in connection with boundary value problems. 4.1.1 H¨ older Spaces Let Ω be a subset of Rn and let 0 < θ < 1. A function ϕ defined on Ω is said to be H¨ older continuous with exponent θ if the quantity [ϕ]θ;Ω = sup

x,y∈Ω x =y

|ϕ(x) − ϕ(y)| |x − y|θ

is finite. We say that ϕ is locally H¨ older continuous with exponent θ if it is H¨older continuous with exponent θ on compact subsets of D. H¨older continuity may be viewed as a fractional differentiability. Let Ω be an open subset of Rn and 0 < θ < 1. We let C θ (Ω) = the space of functions in C(Ω) which are locally H¨older continuous with exponent θ on Ω. If k is a positive integer, we let

106

4 Lp Theory of Pseudo-Differential Operators

C k+θ (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are locally H¨ older continuous with exponent θ on Ω. If K is a compact subset of Ω, we define a seminorm qK,k on C k+θ (Ω) by the formula C k+θ (Ω)  ϕ −→ qK,k (ϕ) = sup |∂ α ϕ(x)| + sup [∂ α ϕ]θ;K . |α|=k

x∈K |α|≤k

It is easy to see that the H¨older space C k+θ (Ω) is a Fr´echet space. Furthermore, we let C θ (Ω) = the space of functions in C(Ω) which are H¨older continuous with exponent θ on Ω. If k is a positive integer, we let C k+θ (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are H¨ older continuous with exponent θ on Ω. Let m be a non-negative integer. We equip the space C m+θ (Ω) with the topology defined by the family {qK,k } of seminorms where K ranges over all compact subsets of Ω. It is easy to see that the H¨older space C m+θ (Ω) is a Fr´echet space. If Ω is a bounded domain of Rn , then C m+θ (Ω) is a Banach space with the norm

ϕ C m+θ (Ω) = ϕ C m (Ω) + sup [∂ α ϕ]θ;Ω |α|=m

= sup |∂ α ϕ(x)| + sup [∂ α ϕ]θ;Ω . x∈Ω |α|≤m

|α|=m

4.1.2 Lp Spaces Throughout this subsection, Ω will denote a bounded domain in Euclidean space Rn . By a measurable function on Ω, we shall mean an equivalence class of measurable functions on Ω which differ only on a subset of measure zero. Any pointwise property attributed to a measurable function will thus be understood to hold true in the usual sense for some function in the same equivalence class. The supremum and infimum of a measurable function will then be understood as the essential supremum and infimum. First, if 1 ≤ p < ∞, we let Lp (Ω) = the space of (equivalence classes of) Lebesgue

4.1 Function Spaces

107

measurable functions u(x) on Ω such that |u(x)|p is integrable on Ω. The space Lp (Ω) is a Banach space with the norm 

u p =

1/p |u(x)| dx . p

Ω

For p = ∞, we let L∞ (Ω) = the space of (equivalence classes of) essentially bounded, Lebesgue measurable functions u(x) on Ω. The space L∞ (Ω) is a Banach space with the norm

u ∞ = ess supx∈Ω |u(x)|. 4.1.3 Fourier Transforms If f ∈ L1 (Rn ), we define its (direct) Fourier transform f by the formula   f (ξ) = e−i x·ξ f (x) dx for ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn , (4.1) Rn

where x · ξ = x1 ξ1 + x2 ξ2 + . . . + xn ξn . It follows from an application of the Lebesgue dominated convergence theorem ([42, Theorem 2.24]) that the function f(ξ) is continuous on Rn , and further we have the inequality

f ∞ = sup |f(ξ)| ≤ f 1 . ξ∈Rn

We also denote f by F f . Similarly, if g ∈ L1 (Rn ), we define the function gˇ(x) by the formula  1 gˇ(x) = ei x·ξ g(ξ) dξ for x = (x1 , x2 , . . . , xn ) ∈ Rn . (2π)n Rn The function gˇ(x) is called the inverse Fourier transform of g. We also denote gˇ by F ∗ g. Now we introduce a subspace of L1 (Rn ) which is invariant under the Fourier transform. We let S(Rn ) = the space of smooth functions ϕ(x) on Rn such that, for any non-negative integer j, the quantity   pj (ϕ) = sup (1 + |x|2 )j/2 |∂ α ϕ(x)| x∈Rn |α|≤j

108

4 Lp Theory of Pseudo-Differential Operators

is finite. The space S(Rn ) is called the Schwartz space or space of smooth functions on Rn rapidly decreasing at infinity. We equip the space S(Rn ) with the topology defined by the countable family {pj } of seminorms. It is easy to verify that S(Rn ) is complete; so it is a Fr´echet space. Now we give typical examples of functions in S(R): Example 4.1. (1) For every a > 0, we let ϕa (x) = e−a

2

x2

ψa (x) = x2 e−a

2

∈ S(R), x2

∈ S(R).

The Fourier transform ϕ a (ξ) of ϕa (x) is given by the formula  ϕ a (ξ) =

e

−i x·ξ −a2 x2

e

R

ξ2 √ π − 2 e 4a dx = a

for ξ ∈ R.

Hence the Fourier transform ψa (ξ) of ψa (x) is given by the formula  2 2 ψa (ξ) = e−i x·ξ x2 e−a x dx R   2 2 ∂2 a ∂2ϕ =− 2 e−i x·ξ e−a x dx = − 2 ∂ξ ∂ξ R ξ2  √  − π ξ2 = 3 1 − 2 e 4a2 for ξ ∈ R. 2a 2a -t (ξ) of the heat kernel (2) The Fourier transform K x2 − 1 e 4t ∈ S(R) Kt (x) = √ 4πt

for x ∈ Rn and t > 0,

is given by the formula -t (ξ) = √ 1 K 4πt = e−t ξ

2

 e R

−i x·ξ

x2 1 e 4t dx = √ ϕ 1/(2√t) (ξ) 4πt −

for ξ ∈ R and t > 0.

The next theorem summarizes the basic properties of the Fourier transform ([42, Section 8.3], [146, Section 1.9]): Theorem 4.1. (i) The Fourier transforms F and F ∗ map S(Rn ) continuously into itself. Furthermore, we have, for all multi-indices α and β,

4.1 Function Spaces α ϕ(ξ) = ξ α ϕ(ξ)   D

109

for every ϕ ∈ S(Rn ),

 β ϕ(ξ)  = (−x) Dβ ϕ(ξ)

for every ϕ ∈ S(Rn ).

(ii) The Fourier transforms F and F ∗ are isomorphisms of S(Rn ) onto itself; more precisely, F F ∗ = F ∗ F = I on S(Rn ). In particular, we have the formula  1 ei x·ξ ϕ(ξ)  dξ for every ϕ ∈ S(Rn ). (4.2) ϕ(x) = (2π)n Rn Formula (4.2) is called the Fourier inversion formula (iii) If ϕ, ψ ∈ S(Rn ), we have the formulas    ϕ(x)ψ(x) dx = ϕ(ξ)ψ(ξ)  dξ, n Rn R  1 ϕ(x)ψ(x) dx = ϕ(ξ)ψ(ξ) dξ. n (2π) n R Rn

(4.3a) (4.3b)

Formulas (4.3a) and (4.3b) are called the Parseval formulas. 4.1.4 Tempered Distributions For the spaces C0∞ (Rn ), S(Rn ) and C ∞ (Rn ), we have the following two inclusions (i) and (ii): (i) The injection of C0∞ (Rn ) into S(Rn ) is continuous and the space C0∞ (Rn ) is dense in S(Rn ). (ii) The injection of S(Rn ) into C ∞ (Rn ) is continuous and the space S(Rn ) is dense in C ∞ (Rn ). For any given function ϕ ∈ S(Rn ) (resp. ϕ ∈ C ∞ (Rn )), it is easy to verify that ψj ϕ −→ ϕ in S(Rn ) (resp. in C ∞ (Rn )) as j → ∞. Hence the dual space S  (Rn ) = L(S(Rn ), C) can be identified with a linear subspace of D (Rn ) containing E  (Rn ), by the identification of a continuous linear functional on S(Rn ) with its restriction to C0∞ (Rn ). Namely, we have the inclusions E  (Rn ) ⊂ S  (Rn ) ⊂ D (Rn ). The elements of S  (Rn ) are called tempered distributions on Rn . In other words, the tempered distributions are precisely those distributions on Rn that have continuous extensions to S(Rn ). Roughly speaking, the tempered distributions are those which grow at most polynomially at infinity, since the functions in S(Rn ) die out faster than any power of x at infinity. In fact, we have the following four examples (1) through (4) of tempered distributions:

110

4 Lp Theory of Pseudo-Differential Operators

(1) The functions in Lp (Rn ) (1 ≤ p ≤ ∞) are tempered distributions. (2) A locally integrable function on Rn is a tempered distribution if it grows at most polynomially at infinity. (3) If u ∈ S  (Rn ) and f (x) is a smooth function on Rn all of whose derivatives grow at most polynomially at infinity, then the product f u is a tempered distribution. (4) Any derivative of a tempered distribution is also a tempered distribution. Now we give some concrete and important examples of distributions in the space S  (Rn ): Example 4.2. (a) The Dirac measure: δ(x). (b) Riesz potentials: Rα (x) =

∂Ω((n − α)/2) 1 n−α α n/2 2 π Γ (α/2) |x|

for 0 < α < n.

(c) Newtonian potentials: N (x) =

Γ ((n − 2)/2) 1 |x|n−2 4 π n/2

for n ≥ 3.

(d) Riesz kernels: Rj (x) =

√ Γ ((n + 1)/2) xj −1 v. p. n+1 |x| π (n+1)/2

for 1 ≤ j ≤ n.

The distribution v. p. (xj /|x|n+1 ) is an extension of v. p. (1/x) to the ndimensional case. The importance of tempered distributions lies in the fact that they have Fourier transforms. If u ∈ S  (Rn ), we define its (direct) Fourier transform F u = u  by the formula (4.4) F u, ϕ = u, F ϕ for all ϕ ∈ S(Rn ). Then we have F u ∈ S  (Rn ), since the Fourier transform F : S(Rn ) −→ S(Rn ) is an isomorphism. Furthermore, in view of formulas (4.3a) and (4.3b) it follows that the above definition (4.4) agrees with definition (4.1) if u ∈ S(Rn ). We also denote F u by u . Similarly, if v ∈ S  (Rn ), we define its inverse Fourier transform F ∗ v = vˇ by the formula F ∗ v, ψ = v, F ∗ ψ

for all ψ ∈ S(Rn ).

The next theorem, which is a consequence of Theorem 4.1, summarizes the basic properties of Fourier transforms in the space S  (Rn ) (see [26, Chapitre I, Th´eor`eme 2.10]):

4.1 Function Spaces

111

Theorem 4.2. (i) The Fourier transforms F and F ∗ map S  (Rn ) continuously into itself. Furthermore, we have, for all multi-indices α and β, F (Dα u)(ξ) = ξ α F u(ξ)

for every u ∈ S  (Rn ),

Dξβ (F u(ξ)) = F ((−x)β u)(ξ)

for every u ∈ S  (Rn ).

(ii) The Fourier transforms F and F ∗ are isomorphisms of S  (Rn ) onto itself; more precisely, F F ∗ = F ∗ F = I on S  (Rn ). (iii) The transforms F and F ∗ are norm-preserving operators on L2 (Rn ) and F F ∗ = F ∗ F = I on L2 (Rn ). This assertion is referred to as the Plancherel theorem. The situation of Theorems 4.1 and 4.2 can be visualized as in Figures 4.3 and 4.4 below. F

S (Rn ) − −−−− → S (Rn )

F

−−−− → L2 (Rn ) L2 (Rn ) −

−−−− → S(Rn ) S(Rn ) − F

Fig. 4.3. The mapping properties of the Fourier transform F

F∗

S (Rn ) ←−−−− − S (Rn )

F∗

L2 (Rn ) ←−−−− − L2 (Rn )

− − S(Rn ) S(Rn ) ←−−− ∗ F

Fig. 4.4. The mapping properties of the inverse Fourier transform F ∗

Finally, we give a typical example of tempered distributions from the viewpoint of distribution kernels in the theory of pseudo-differential operators:

112

4 Lp Theory of Pseudo-Differential Operators

Example 4.3. Let 0 < α < 2. Then we can define a tempered distribution 1 v.p. n+α as follows: |x| (1) The case where 0 < α < 1: . /  1 ϕ(y) − ϕ(0) v.p. n+α , ϕ = lim dy ε↓0 |y|≥ε |x| |y|n+α for all ϕ ∈ S(Rn ). (2) The case where 1 ≤ α < 2: . /  ϕ(y) + ϕ(−y) − 2ϕ(0) 1 v.p. n+α , ϕ = lim dy ε↓0 |y|≥ε |x| |y|n+α for all ϕ ∈ S(Rn ). Here “v.p.” stands for Cauchy’s “valeur principale” in French. Moreover, its Fourier transform is given by the formula (see [122, Example 5.28])  F v.p.

1 |x|n+α

 =

π n/2 Γ (−α/2) |ξ|α . 2α Γ ((α + n)/2)

4.1.5 Sobolev Spaces If s ∈ R, we define a linear map J s : S  (Rn ) −→ S  (Rn ) by the formula J su = F ∗

  −s/2 1 + |ξ|2 Fu

for all u ∈ S  (Rn ).

This formula can be visualized as in Figure 4.5 below. J s =(1−Δ)−s/2

u ∈ S (Rn ) −−−−−−−−−−→

S (Rn )

J su F∗

F

F u ∈ S (Rn ) −−−−−−−−→ S (Rn ) (1+|ξ|2 )−s/2

(1 + |ξ|2 )−s/2 F u

Fig. 4.5. The definition of the Bessel potential J s

Then it is easy to see that the map J s is an isomorphism of S  (Rn ) onto itself and that its inverse is the map J −s . The function J s u is called the Bessel potential of order s of u.

4.1 Function Spaces

113

It should be noticed that if we define a function Gs (x) by the formula 1 1 Gs (x) := Γ (s/2) (4π)n/2



∞

e 0

−t−

|x|2 s − n dt 4t t 2 t

for s > 0,

then we have the important formula (see Stein [108, Chapter V, Section 3])   -s (ξ) = 1 + |ξ|2 −s/2 G

for s > 0.

It is known (see Aronszajn–Smith [12]) that the function Gs (x) is represented in the form Gs (x) =

1 K(n−s)/2 (|x|) |x| 2(n+s−2)/2 π n/2 Γ (s/2)

s−n 2 ,

where K(n−s)/2 (z) is the modified Bessel function of the third kind (see Watson [142]). First, we define Sobolev spaces of fractional order or Bessel potential spaces (see Figure 4.6): Definition 4.3. If s ∈ R and 1 < p < ∞, we let H s,p (Rn ) = the image of Lp (Rn ) under the mapping J s . We equip H s,p (Rn ) with the norm  

u s,p = J −s up

for u ∈ H s,p (Rn ).

The space H s,p (Rn ) is called the Sobolev space of fractional order s or Bessel potential space of order s. The situation can be visualized as in Figure 4.6 below. Js

u = J s v ∈ H s,p (Rn ) ←− Lp (Rn )

v = J −s u

Fig. 4.6. The Sobolev spaces H s,p (Rn ) and Lp (Rn )

We list some basic topological properties of H s,p (Rn ) ([2], [135]): (1) The Schwartz space S(Rn ) is dense in each H s,p (Rn ).  (2) The space H −s,p (Rn ) is the dual space of H s,p (Rn ), where p = p/(p − 1) is the exponent conjugate to p ([2, p. 62, Theorem 3.9], [135, p. 178, Theorem 2.11.2]).

114

4 Lp Theory of Pseudo-Differential Operators

(3) If s > t, then we have the inclusions S(Rn ) ⊂ H s,p (Rn ) ⊂ H t,p (Rn ) ⊂ S  (Rn ), with continuous injections. (4) If s is a non-negative integer, then the space H s,p (Rn ) is isomorphic to the usual Sobolev space W s,p (Rn ), that is, the space H s,p (Rn ) coincides with the space of functions u ∈ Lp (Rn ) such that Dα u ∈ Lp (Rn ) for |α| ≤ s, and the norm · s,p is equivalent to the norm ⎛ ⎝

  |α|≤s

Rn

⎞1/p |Dα u(x)|p dx⎠

.

4.1.6 Besov Spaces Secondly, we define Besov spaces: Definition 4.4. (1) If 1 < p < ∞, we let B 1,p (Rn−1 ) = the space of (equivalence classes of ) functions ϕ(x ) ∈ Lp (Rn−1 ) for which the integral  |ϕ(x + y  ) − 2ϕ(x ) + ϕ(x − y  )|p   dy dx |y  |n−1+p Rn−1 ×Rn−1 is finite. The space B 1,p (Rn−1 ) is a Banach space with respect to the norm  |ϕ(x )|p dx |ϕ|1,p = Rn−1

 +

Rn−1 ×Rn−1

|ϕ(x + y  ) − 2ϕ(x ) + ϕ(x − y  )|p   dy dx |y  |n−1+p

(2) If p = ∞, we let B 1,∞ (Rn−1 ) = the space of (equivalence classes of ) functions ϕ(x ) ∈ L∞ (Rn−1 ) for which the quantity

ϕ(· + y  ) − 2ϕ(·) + ϕ(· − y  ) ∞ |y  | |y  |>0 sup

is finite. The space B 1,∞ (Rn−1 ) is a Banach space with respect to the norm

ϕ(· + y  ) − 2ϕ(·) + ϕ(· − y  ) ∞ . |y  | |y  |>0

|ϕ|1,∞ = ϕ ∞ + sup

1/p .

4.1 Function Spaces

115

(3) If s ∈ R and 1 < p ≤ ∞, we let B s,p (Rn−1 ) = the image of B 1,p (Rn−1 ) under the mapping J where J  is the Bessel potential of order s − 1 on Rn−1 . s−1

s−1

We equip the space B s,p (Rn−1 ) with the norm    −s+1  ϕ for ϕ(x ) ∈ B s,p (Rn−1 ). |ϕ|s,p = J  1,p

The space B s,p (Rn−1 ) is called the Besov space of order s. The situation can be visualized as in Figure 4.7 below.

ϕ=J

s−1

J s−1

ψ ∈ B s,p (Rn−1 ) ←− B 1,p (Rn−1 )

ψ=J

−s+1

ϕ

Fig. 4.7. The Besov spaces B s,p (Rn−1 ) and B 1,p (Rn−1 )

We list some basic topological properties of B s,p (Rn−1 ) (see [135]): (1) The Schwartz space S(Rn−1 ) is dense in each B s,p (Rn−1 ).  (2) The space B −s,p (Rn−1 ) is the dual space of B s,p (Rn−1 ), where p = p/(p − 1) is the exponent conjugate to p ([135, p. 178, Theorem 2.11.2]). (3) If s > t, then we have the inclusions S(Rn−1 ) ⊂ B s,p (Rn−1 ) ⊂ B t,p (Rn−1 ) ⊂ S  (Rn−1 ), with continuous injections. (4) If s = m + σ where m is a non-negative integer and 0 < σ < 1, then the Besov space B s,p (Rn−1 ) coincides with the space of functions ϕ(x ) ∈ H m,p (Rn−1 ) such that, for |α| = m the integral (Slobodecki˘ı seminorm)  |Dα ϕ(x ) − Dα ϕ(y  )|p   dx dy < ∞. |x − y  |n−1+pσ Rn−1 ×Rn−1 Furthermore, the norm |ϕ|s,p is equivalent to the norm ([135, p. 36, part (iv)])    |Dα ϕ(x )|p dx |α|≤m

+

Rn−1

 

|α|=m

Rn−1 ×Rn−1

|Dα ϕ(x ) − Dα ϕ(y  )|p   dx dy |x − y  |n−1+pσ

1/p .

116

4 Lp Theory of Pseudo-Differential Operators

4.1.7 General Sobolev and Besov Spaces Now we define the Sobolev spaces H s,p (Ω) and H s,p (M ) of fractional order and the Besov spaces B s,p (∂Ω) for a bounded domain Ω of Rn with smooth boundary ∂Ω and the double M of Ω (see Figure 4.1): Definition 4.5. If s ∈ R and 1 < p < ∞, we define H s,p (Ω) = the space of distributions u ∈ D (Ω) such that there exists a function U ∈ H s,p (Rn ) with U |Ω = u, and equip the space H s,p (Ω) with the norm

u s,p = inf U s,p , where the infimum is taken over all such U . The space H s,p (Ω) is a Banach space with respect to the norm · s,p . We remark that H 0,p (Ω) = Lp (Ω);

· 0,p = · p .

Then we have the following important relationships between the Lp Sobolev spaces H s,p (Ω) and W s,p (Ω) for all s ≥ 0 and 1 < p < ∞ (see [81, Theorem]): Theorem 4.6 (Meyers–Serrin). If Ω is a bounded, Lipschitz domain, then we have, for all s ≥ 0 and 1 < p < ∞, H s,p (Ω) = W s,p (Ω). We introduce a space of distributions on Ω which behave locally just like the distributions in H s,p (Rn ): Definition 4.7. (1) If s ∈ R and 1 < p < ∞, we define s,p Hloc (Ω) = the space of distributions u ∈ D (Ω) such that ϕu ∈ H s,p (Rn ) for all ϕ ∈ C0∞ (Ω). s,p We equip the localized Sobolev space Hloc (Ω) with the topology defined by the seminorms u −→ ϕu s,p s,p as ϕ ranges over C0∞ (Ω). It is easy to verify that Hloc (Ω) is a Fr´echet space. s,p (2) Similarly, the localized Besov space Bloc (∂Ω) is defined for s ∈ R and 1 < p ≤ ∞, with H s,p (Rn ) replaced by B s,p (Rn−1 ).

If M is the double of Ω, then the Sobolev spaces H s,p (M ) of fractional order are defined to be locally the spaces H s,p (Rn ), upon using local coordinate systems flattening out M , together with a partition of unity. Similarly, the Besov spaces B s,p (∂Ω) are defined with H s,p (Rn ) replaced by B s,p (Rn−1 ). The norms of H s,p (M ) and B s,p (∂Ω) will be denoted by · s,p and | · |s,p , respectively.

4.1 Function Spaces

117

4.1.8 Sobolev’s Imbedding Theorems It is the imbedding characteristics of Lp Sobolev spaces that render these spaces so useful in the study of partial differential equations. In the proof of Theorem 1.5 (see [121, Theorem 1.3]) we need the following imbedding properties of Lp Sobolev spaces (see the proof of Lemma 12.5): Theorem 4.8 (Sobolev). Let Ω be a bounded domain in the Euclidean space Rn with boundary ∂Ω of class C 2 . Let j and m be non-negative integers and 1 ≤ p < ∞. Then we have the following three assertions (i), (ii) and (iii): (i) If 1 ≤ p < n, we have the continuous injection H 2,p (Ω) ⊂ H 1,q (Ω)

for

1 1 1 1 − ≤ ≤ . p n q p

(ii) If (m − 1)p < n < mp, we have the continuous injection H j+m,p (Ω) ⊂ C j+ν (Ω)

for 0 < ν ≤ m −

n . p

(iii) If (m − 1)p = n, we have the continuous injection H j+m,p (Ω) ⊂ C j+ν (Ω)

for 0 < ν < 1.

For a proof of Theorem 4.8, see Adams–Fournier [2, Theorem 4.12], Friedman [43, Part I, Theorems 10.2 and 11.1] and also [123, Theorems 4.17 and 4.19]. Since the elements of H k,p (Ω) are, strictly speaking, not functions defined everywhere in Ω but rather equivalence classes of such functions defined and equal up to sets of measure zero, we must clarify what it meant by an imbedding of H j+m,p (Ω) into C j+ν (Ω). The imbedding H j+m,p (Ω) → C j+ν (Ω) means that each u ∈ H j+m,ν (Ω) can, when considered as a function, be redefined on a set of zero measure in Ω in such a way that the modified function u 0 (which equals u in H j+m,ν (Ω)) belongs to C j+ν (Ω) and satisfies the inequality

0 u C j+ν (Ω) ≤ C u H j+m,ν (Ω) , with some constant C > 0 depending on m, p, n and j. In the proof of Theorem 1.5 (see [121, Theorem 1.3]) we make use of the following inequality (see the proof of Lemma 12.1): Theorem 4.9 (Gagliardo–Nirenberg). Let Ω be a bounded domain in Rn with boundary of class C 2 , and 1 ≤ p, r ≤ ∞. Then we have the following two assertions (i) and (ii): (i) If p = n and if the inequality   1 1 2 1 1 = +θ − + (1 − θ) q n p n r

for

1 ≤θ≤1 2

118

4 Lp Theory of Pseudo-Differential Operators

holds true, then we have, for all functions u ∈ H 2,p (Ω) ∩ Lr (Ω),

u 1,q ≤ C1 u θ2,p u 1−θ , r

(4.5)

with a positive constant C1 = C1 (Ω, p, r, θ). (ii) If n/2 < p < ∞, p = n and if the inequality   n n 0≤ν t, then the injections H s,p (Ω) −→ H t,p (Ω), B s,p (∂Ω) −→ B t,p (∂Ω) are both compact (or completely continuous). The Rellich–Kondrachov theorem is an Lp Sobolev space version of the Bolzano–Weierstrass theorem and the Ascoli–Arzel`a theorem in calculus (see Table 4.1 below): For a proof of Theorem 4.10, see Adams–Fournier [2, Theorem 6.3 and Paragraph 7.32], Friedman [43, Part I, Theorem 11.2], Gilbarg–Trudinger [50, Theorem 7.22] and also Triebel [135, p. 233, Remark 1].

4.2 Seeley’s Extension Theorem The next theorem, due to Seeley [103], asserts that the functions in C ∞ (Ω) are the restrictions to Ω of functions in C ∞ (Rn ) (see [103], [2, Theorems 5.21 and 5.22]):

4.2 Seeley’s Extension Theorem

Subjects

Sequences

Compactness theorems

Theory of real numbers

Sequences of real numbers

The Bolzano–Weierstrass theorem

Calculus

Sequences of continuous functions

The Ascoli–Arzer` a theorem

Theory of distributions

Sequences of distributions

The Rellich–Kondrachov theorem

119

Table 4.1. A bird’s-eye view of three compactness theorems in calculus

Theorem 4.11 (Seeley’s extension theorem). Let Ω be either the half space Rn+ or a smooth domain in Rn with bounded boundary ∂Ω. Then there exists a continuous linear extension operator E : C ∞ (Ω) −→ C ∞ (Rn ). Furthermore, the restriction to the space C0∞ (Ω) of E is a continuous linear extension operator on C0∞ (Ω) into C0∞ (Rn ). Here C0∞ (Ω) = the space of restrictions to Ω of functions in C0∞ (Rn ). Proof. The proof of Theorem 4.11 is divided into Steps I and II. Step I: First, we consider the case where Ω = Rn+ . The proof is based on the following elementary lemma (see [123, Lemma 4.22]): Lemma 4.12. There exists a function w(t) in the space S(R) such that  supp w ⊂ [1, ∞), +∞ n n n = 0, 1, 2, . . . . 1 t w(t) dt = (−1) , In this book, assuming Lemma 4.12 we prove Theorem 4.11. By using the function w in Lemma 4.12, we can define a linear operator E : C ∞ (Rn+ ) −→ C ∞ (Rn ) by the formula

120

4 Lp Theory of Pseudo-Differential Operators

 

Eϕ(x , xn ) = where and

ϕ(x , xn ) +∞ w(s) θ(−xn s) ϕ(x , −sxn ) ds 1

x = (x , xn ) ∈ Rn ,

if xn ≥ 0, if xn < 0,

x = (x1 , x2 , . . . , xn−1 ) ∈ Rn−1 ,

⎧ ∞ ⎪ ⎨θ ∈ C0 (R), supp θ ⊂ [−2, 2], ⎪ ⎩ θ(t) = 1 for |t| ≤ 1.

Then it is easy to verify the following three assertions (1), (2) and (3): (1) Eϕ ∈ C ∞ (Rn ). (2) The operator E maps C ∞ (Rn+ ) continuously into C ∞ (Rn ). (3) If supp ϕ ⊂ {x ∈ Rn : |x | ≤ r, 0 ≤ xn ≤ a} for some r > 0 and a > 0, then it follows that supp Eϕ ⊂ {x ∈ Rn : |x | ≤ r, |xn | ≤ a}. This proves Theorem 4.11 for the half space Rn+ . Step II: Now we assume that Ω is a smooth domain in Rn with bounded boundary ∂Ω. Then we can choose a finite covering {Vj }N j=1 of ∂Ω by open subsets of Rn and C ∞ diffeomorphisms χj of Vj onto the unit ball B(0, 1) = {x ∈ Rn : |x| < 1} (see Figures 4.8 and 4.9 below) such that the open sets $ √ '% 1 3 −1 n  Vj = χj , x ∈ R : |x | < , |xn | < 2 2

1 ≤ j ≤ N,

form an open covering of the tubular neighborhood Ωδ = {x ∈ Ω : dist (x, ∂Ω) < δ} for some δ > 0 . Furthermore, we can choose an open set V0 in Ω, bounded away from ∂Ω (see Figure 4.10 below) such that   Ω ⊂ V0 ∪ ∪N j=1 Vj . Let {ωj }N j=0 be a partition of unity subordinate to the open covering {Vj }N . Namely, the family {ωj }N j=0 j=0 satisfies the following three conditions (PU1), (PU2) and (PU3): (PU1) 0 ≤ ωj (x) ≤ 1 for all x ∈ Ω and 0 ≤ j ≤ N . (PU2) supp ωj ⊂ Vj for each 0 ≤ j ≤ N . (PU3) The functions {ωj }N j=0 satisfy the condition N  j=0

ωj (x) = 1 for every x ∈ Ω.

4.2 Seeley’s Extension Theorem

121

∂Ω Ω Vj Vi ωj ωi

Fig. 4.8. The open covering {Vj } and the partition of unity {ωj } xn

∂Ω Ω

B(0, 1)

Vj

χ

x = (x1 , . . . , xn−1 )

Fig. 4.9. The coordinate transformation χj maps Vj onto B(0, 1)

Ω Vj V0

Fig. 4.10. The open sets V0 and Vj

If ϕ ∈ C ∞ (Ω), we define a linear operator Eϕ = ω0 ϕ +

N 

   ∗ χ∗j E (χ−1 . j ) (ωj ϕ)

j=1

Then it is easy to verify that this operator E enjoys the desired properties. Theorem 4.11 is proved, apart from the proof of Lemma 4.12.  

122

4 Lp Theory of Pseudo-Differential Operators

4.3 Trace Theorems First, we study the restrictions to the hyperplane {xn = 0} of functions in the Lp Sobolev space H s,p (Rn+ ) on the half-space Rn+ . If x = (x1 , . . . , xn ) is a point of Rn , we write x = (x , xn ),

x = (x1 , . . . , xn−1 ) ∈ Rn−1 .

If j is a non-negative integer, we define the trace map γj : C0∞ (Rn+ ) −→ C0∞ (Rn−1 ) by the formula ∞ γj u(x ) = lim Dnj u(x , xn ) for u ∈ C(0) (Rn+ ). xn ↓0

Then we have the following theorem: Theorem 4.13. Let 1 < p < ∞. If 0 ≤ j < s − 1/p, then the trace map ∞ γj : C(0) (Rn+ ) −→ C0∞ (Rn−1 )

extends uniquely to a continuous linear map γj : H s,p (Rn+ ) −→ B s−j−1/p,p (Rn−1 ). Furthermore, if u ∈ H s,p (Rn+ ), then the mapping xn −→ Dnj u(·, xn ) is a continuous function on [0, ∞) with values in B s−j−1/p,p (Rn−1 ). The next theorem shows that the result of Theorem 4.13 is sharp: Theorem 4.14 (the trace theorem). Let 1 < p < ∞. If 0 ≤ j < s − 1/p, then the trace map 1 γ : H s,p (Rn+ ) −→ B s−j−1/p,p (Rn−1 ) 0≤j 0. Remark 4.20. Theorem 4.19 is an expression of the fact that if we know about the derivatives of the solution u of Au = f in tangential directions, then we can derive information about the normal derivatives γj u by means of the equation Au = f . Proof. By using local coordinate systems flattening out the boundary ∂Ω, together with a partition of unity, we may assume that Ω = Rn+ , u ∈ E  (Rn ), and that u=0

in Rn+ .

The proof of Theorem 4.19 is divided into seven steps.

4 Lp Theory of Pseudo-Differential Operators

126

Step (1): First, we can find a constant C > 0 and a integer  ≥ 0 such that  for all ξ ∈ Rn . | u(ξ)| ≤ C (1 + |ξ|) Hence we have, for every ε > 0, u ∈ H −−n/2−ε (Rn ). n If B ∈ Lm cl (R ) for m < 0, it follows that

Bu ∈ H −−m−n/2−ε (Rn ). By Sobolev’s imbedding theorem, we obtain that Bu ∈ C k (Rn ) for k < −m − n − .

(4.8)

Take a function χ(ξ) ∈ C ∞ (Rn ) such that  0 in a neighborhood of ξ = 0, χ(ξ) = 1 for |ξ| ≥ 1. μ If a(x, ξ) ∈ Scl (Rn × Rn ) is a symbol of A ∈ Lμcl (Rn ), we can express it as follows: a(x, ξ) = χ(ξ) a(x, ξ) + (1 − χ(ξ)) a(x, ξ),

where

(1 − χ(ξ)) a(x, ξ) ∈ S −∞ (Rn × Rn ).

By applying assertion (4.8) with m := −∞, we find that (I − χ(D)) a(x, D)u ∈ C ∞ (Rn ). Hence we are reduced to the following case: ⎧ 0 a(x, D) ∈ Lμ (Rn ), ⎪ ⎨A = 0 cl 0 a(x, ξ) = χ(ξ) a(x, ξ), ⎪ ⎩ 0 a(x, t ξ) = tμ a(x, ξ) for all t > 0. Step (2): Let Φ(x) be an arbitrary function in C0∞ (Rn ). Then we have the formula    2 3 1 i x·ξ 0 Au, Φ = u (ξ) e 0 a(x, ξ) Φ(x) dx dξ (2π)n Rn Rn    1 i x·ξ = u (ξ) e a(x, ξ) Φ(x) dx dξ (2π)n |ξ|≥1 Rn    1 i x·ξ u  (ξ) e χ(ξ) a(x, ξ) Φ(x) dx dξ + (2π)n |ξ|≤1 Rn

4.3 Trace Theorems

127

:= Bu, Φ + Ru, Φ . However, by using Fubini’s theorem we can write down the last term as follows:    1 i x·ξ Ru, Φ = u (ξ) e χ(ξ) a(x, ξ) Φ(x) dx dξ (2π)n |ξ|≤1 Rn %  $ 2 3 1 i(x−y)·ξ e = , u(y) χ(ξ) a(x, ξ) dξ Φ(x) dx (2π)n Rn |ξ|≤1  = KR (x, y), u(y) Φ(x) dx, Rn

where KR (x, y) is the distribution kernel of R, given by the formula  1 KR (x, y) = ei(x−y)·ξ χ(ξ) a(x, ξ) dξ. (2π)n |ξ|≤1 Since we have the assertion KR (x, y) ∈ C ∞ (Rn × Rn ), it follows that

Ru ∈ C ∞ (Rn ).

Therefore, we are reduced to the study of the term 0 − Ru, Bu := Au where

 1 u (ξ)F (ξ) dξ, Bu, Φ = (2π)n |ξ|≥1  F (ξ) = ei x·ξ a(x, ξ) Φ(x) dx. Rn

Step (3): Since a(x, ξ) is a rational function of ξ, we find that the poles of the function a(x, ξ  , ξn ) of ξn remains in some compact subset of the complex place C when x belongs to a compact subset of Rn and ξ  belongs to a compact subset of Rn−1 , respectively. Hence, for every fixed ξ  ∈ Rn−1 the function     ei x ·ξ ei xn ξn a(x, ξ  , ξn ) Φ(x) dx C  ξn −→ F (ξ , ξn ) = Rn

is holomorphic, for |ξn | sufficiently large. Moreover, we have the following claim: Claim 4.21. Assume that Φ(x) ∈ C0∞ (Rn ) satisfies the condition supp Φ ⊂ Rn+ .

128

4 Lp Theory of Pseudo-Differential Operators

Then the function F (ξ  , ξn ) =

 ei x·ξ a(x, ξ) Φ(x) dx Rn

is rapidly decreasing with respect to the variable ξ = (ξ  , ξn ), for ξ  ∈ Rn−1 and Im ξn ≥ 0. Proof. By integration by parts, we have, for any multi-index α,  α ξ F (ξ) = Dxα ei x·ξ · a(x, ξ  , ξn ) Φ(x) dx Rn  ei x·ξ · Dxα (a(x, ξ  , ξn ) Φ(x)) dx. = (−1)|α| Rn

However, we remark that 







• ei x·ξ = ei x ·ξ ei xn ξn = ei x ·ξ ei xn Re ξn · e−xn Im ξn , • x ∈ supp Φ =⇒ xn > 0. Hence we have, for some constant Cα > 0, |ξ α F (ξ)| ≤ C (1 + |ξ|)μ

for all ξ  ∈ Rn−1 and Im ξn ≥ 0.

Therefore, for any multi-index α we can find a constant Cα > 0 such that |F (ξ)| ≤ Cα (1 + |ξ|)

μ−|α|

for all ξ  ∈ Rn−1 and Im ξn ≥ 0.

The proof of Claim 4.21 is complete.   On the other hand, we have the following claim: Claim 4.22. Assume that u = 0 in Rn+ , that is, supp u ⊂ {x = (x , xn ) ∈ Rn : xn ≤ 0} . Then the function u (ξ  , ξn ) is slowly increasing with respect to the variable   ξ = (ξ , ξn ), for ξ ∈ Rn−1 and Im ξn ≥ 0. Proof. Since u ∈ E  (Rn ), we can find a constant C > 0 and a non-negative integer  such that ⎛ ⎞  |u, ϕ| ≤ C sup ⎝ |∂ α ϕ(x)|⎠ for all ϕ ∈ C ∞ (Rn ). (4.9) x∈supp u

Thus, by taking we have the formula

|α|≤

ϕ(x) := e−i x·ξ ,

4.3 Trace Theorems

129

3 5 2 4   u (ξ  , ξn ) = u, e−i x·ξ = u, e−i x ·ξ e−i xn ξn 3 2   = u, e−i x ·ξ e−i xn Re ξn exn Im ξn . However, since u = 0 in Rn+ , it follows that exn Im ξn ≤ 1

for all x ∈ supp u and Im ξn ≥ 0.

Therefore, we have, by inequality (4.9), 

| u(ξ)| ≤ C (1 + |ξ|)

for all x ∈ supp u and Im ξn ≥ 0,

just as in the proof of the Paley–Wiener–Schwartz theorem. The proof of Claim 4.22 is complete.   Step (4): Now we make a contour deformation in the integral  1 Bu, Φ = u (ξ)F (ξ) dξ (2π)n |ξ|≥1  1 u (ξ  , ξn )F (ξ  , ξn ) dξ  dξn . = (2π)n |ξ|≥1 We consider the two cases for |ξ| ≥ 1: (a) |ξ  | ≥ 1 and −∞ 0} enclosing the poles of a(x, ξ  , ξn ) of ξn . Then we have, by Cauchy’s theorem,   R u (ξ  , ξn )F (ξ  , ξn ) dξn + u (ξ  , ξn )F (ξ  , ξn ) dξn −R CR  = u (ξ  , ξn ) F (ξ  , ξn ) dξn . Γξ

However, by Claims 4.21 and 4.22 it follows that  u (ξ  , ξn ) F (ξ  , ξn ) dξn = 0. lim R→∞

CR

Hence, by passing to the limit we obtain that   ∞   u (ξ , ξn ) F (ξ , ξn ) dξn = u (ξ  , ξn )F (ξ  , ξn ) dξn −∞

for all |ξ  ≥ 1.

Γξ

Case (b): In this case, we take a positive contour Γξ in the upper halfplane {Im ξn > 0} enclosing the poles of a(x, ξ  , ξn ) of ξn , completed by the segment

4 Lp Theory of Pseudo-Differential Operators

130

 6  6 2 2 − 1 − |ξ  | , 1 − |ξ  | . Similarly, we have, by Cauchy’s theorem,   R u (ξ  , ξn )F (ξ  , ξn ) dξn + √ 1−|ξ  |2

 + 

−

√

−R

=

1−|ξ  |2

u (ξ  , ξn )F (ξ  , ξn ) dξn

CR

u (ξ  , ξn )F (ξ  , ξn ) dξn

u (ξ  , ξn )F (ξ  , ξn ) dξn .

Γξ

Hence, by passing to the limit we obtain that  ∞  −√1−|ξ |2   u (ξ , ξn )F (ξ , ξn ) dξn + √

1−|ξ  |2

−∞

 =

u (ξ  , ξn )F (ξ  , ξn ) dξn

u (ξ  , ξn )F (ξ  , ξn ) dξn

for all |ξ  < 1.

Γξ

Summing up, we have proved the formula  1 Bu, Φ = u (ξ  , ξn )F (ξ  , ξn ) dξ  dξn (4.10) (2π)n |ξ|≥1  ∞   1   u  (ξ , ξ ) F (ξ , ξ ) dξ = dξ  n n n (2π)n |ξ |≥1 −∞ $ √ %  − 1−|ξ  |2 1 + u (ξ  , ξn ) F (ξ  , ξn ) dξn dξ  (2π)n |ξ | 0, we define

132

4 Lp Theory of Pseudo-Differential Operators

1 x , ρ εn ε function on Rn , and satisfies the conditions

ρε (x) = then ρε (x) is a non-negative, C ∞

supp ρε = {x ∈ Rn : |x| ≤ ε} ;  ρε (x) dx = 1.

(4.13a) (4.13b)

Rn

The functions {ρε } are called Friedrichs’ mollifiers (see Figure 4.12 below).

y

y = ρ 14 (x)

y = ρ 12 (x) y = ρ1 (x) = ρ(x) |x| = 1

0

|x| =

1 4

|x| =

1 2

|x| = 1

Fig. 4.12. Friedrichs’ mollifiers {ρε }

Now we let ρk (x) := k n ρ(kx) for all integer k > 0. Since u ∈ E  (Rn ) satisfies the condition {x = (x , xn ) ∈ Rn : xn ≤ 0} , it follows that ⎧ ∞ n ⎪ ⎨ρk ∗ u ∈ C0 (R ), supp (ρk ∗ u) ⊂ {x = (x , xn ) ∈ Rn : xn ≤ 0} , ⎪ ⎩ ρk ∗ u −→ u in E  (Rn ) as k → ∞.

x ∈ Rn

4.3 Trace Theorems

Hence we have the assertion B (ρk ∗ u) −→ Bu

in D (Rn ) as k → ∞,

and so Bu, ϕ(x ) ⊗ ψ(xn ) = lim B (ρk ∗ u) , ϕ(x ) ⊗ ψ(xn ) k→∞   1 = lim ρk (ξ) u (ξ  , ξn ) k→∞ (2π)n Rn−1 Γξ    i x·ξ     × e a (x , xn , ξ , ξn ) ϕ(x )ψ(xn ) dx dxn dξn dξ  . Rn

Here we remark the following three facts (a), (b) and (c): (a) ρ -k (ξ) = ρ(ξ/k) ∈ S(Rn ) for all integer k > 0. (b) |ξn | ≤ c0 (1 + |ξ  |) for all ξn ∈ Γξ and |ξ  | ≥ 1. (c) Γξ = |ξ  | Γ1 for all |ξ  | ≥ 1. Hence, by using Fubini’s theorem we obtain that   1 ρ -k (ξ  , ξn ) u (ξ  , ξn ) (2π)n Rn−1 Γξ    i x·ξ     × e a (x , xn , ξ , ξn ) ϕ(x ) ψ(xn ) dx dxn dξn dξ  n R       ∞ (ξ , ξn ) 1 i xn ξn = ψ(xn ) e ρ u (ξ  , ξn ) (2π)n 0 k n−1 R Γξ       ei x ·ξ a(x , xn , ξ  , ξn )ϕ(x ) dx dξn dξ  dxn . × Rn−1

However, we have the following assertion (d): (d) The function

 Rn−1





ei x ·ξ a (x , xn , ξ  , ξn ) ϕ(x ) dx

is rapidly decreasing with respect to ξ  , since ϕ(x ) ∈ C0∞ (Rn−1 ). Indeed, by integration by parts it suffices to note that    α ξ ei x ·ξ a(x , xn , ξ  , ξn ) ϕ(x ) dx n−1 R      = Dxα ei x ·ξ a (x , xn , ξ  , ξn ) ϕ(x ) dx Rn−1      |α | ei x ·ξ Dxα (a (x , xn , ξ  , ξn ) ϕ(x )) dx . = (−1) Rn−1

133

4 Lp Theory of Pseudo-Differential Operators

134

Hence, we find that the function     1 (ξ , ξn )  i xn ξn e ρ Gk (xn , ξ ) = u (ξ  , ξn ) 2π Γξ k     × ei x ·ξ a (x , xn , ξ  , ξn ) ϕ(x ) dx dξn Rn−1

is rapidly decreasing with respect to the variable ξ  . Moreover, we have the inequality        ρ (ξ , ξn )  ≤ ρ(0) = ρ(x) dx = 1 for all integer k > 0.   k Rn Therefore, by applying Lebesgue’s dominated convergence theorem we obtain that Bu, ϕ(x ) ⊗ ψ(xn )   1 ρ -k (ξ) u (ξ  , ξn ) = lim k→∞ (2π)n Rn−1 Γξ    i x·ξ     e a (x , xn , ξ , ξn ) ϕ(x ) ψ(xn ) dx dxn dξn dξ  × Rn       ∞ (ξ , ξn ) 1 i xn ξn = lim ψ(xn ) e ρ u (ξ  , ξn ) k→∞ (2π)n 0 k n−1 R Γξ       ei x ·ξ a (x , xn , ξ  , ξn ) ϕ(x ) dx dξn dξ  dxn × Rn−1    ∞ 1 = ψ(x ) ei xn ξn u (ξ  , ξn ) n (2π)n 0 Rn−1 Γξ     i x ·ξ      e a (x , xn , ξ , ξn ) ϕ(x ) dx dξn dξ  dxn × Rn−1    ∞ 1   = ψ(xn ) G(xn , ξ ) dξ dxn , (2π)n−1 0 Rn−1 where G(xn , ξ  )      1 = ei xn ξn u (ξ  , ξn ) ei x ·ξ a (x , xn , ξ  , ξn ) ϕ(x ) dx dξn . 2π Γξ Rn−1 The proof of Claim 4.23 is complete.   Step (7): By virtue of Claim 4.23, we find that the distribution [0, ∞)  xn −→ Bu(·, xn )|Rn

+

4.3 Trace Theorems

135

  is equal to a function in the space C ∞ [0, ∞); D (Rn−1 ) , given by the formula   1 ∞ n−1 C0 (R )  ϕ −→ ei xn ξn u (ξ  , ξn ) (2π)n−1 Rn−1 Γξ      × ei x ·ξ a (x , xn , ξ  , ξn ) ϕ(x ) dx dξn dξ  . Rn−1

Now the proof of Theorem 4.19 is complete.   Remark 4.24. It is easy to see that the function G(xn , ξ  ) is rapidly decreasing with respect to the variable ξ  . 4.3.2 Jump Formulas If u ∈ D (Ω) has a sectional trace on ∂Ω of order zero, we can define its extension u0 in D (M ) as follows: Choose functions θ ∈ C0∞ (W ) and ψ ∈ C0∞ (Ω) such that θ + ψ = 1 on Ω, and define u0 by the formula  1 4 0 5 u ,ϕ ·μ = u(t), (θϕ)(·, t) · ω dt + u, ψϕ · μ , ϕ ∈ C ∞ (M ). 0

0

The distribution u is an extension to M of u which is equal to zero in M \ Ω. If v ∈ D (∂Ω), we define a multiple layer v ⊗ Dtj δ∂Ω

for j = 0, 1, . . .,

by the formula 2 2 3 3 v ⊗ Dtj δ∂Ω , ϕ · μ = (−1)j v, Dtj ϕ(·, 0) · ω

for all ϕ ∈ C ∞ (M ).

It is clear that v ⊗ Dtj δ is a distribution on M with support in ∂Ω. Let P be a differential operator of order m with C ∞ coefficients on M . In a neighborhood of ∂Ω, we can write uniquely P = P (x, Dx ) in the form P (x, Dx ) =

m 

Pj (x, Dx )Dtj ,

x = (x , t),

j=0

where Pj (x, Dx ) is a differential operator of order m − j acting along the surfaces parallel to the boundary ∂Ω. Then we have the following two formulas (1) and (2): (1) If u ∈ D (Ω) has sectional traces on ∂Ω up to order j, then we have the formula j−1 1  Dtj (u0 ) = (Dtj u)0 + √ γj−k−1 u ⊗ Dtk δ∂Ω . (4.14) −1 k=0 (2) If u ∈ D (Ω) has sectional traces on ∂Ω up to order m, then we have the jump formula    1 0 P+k+1 (x, Dx )γ u ⊗ Dtk δ∂Ω . (4.15) P u0 = (P u) + √ −1 +k+1≤m

136

4 Lp Theory of Pseudo-Differential Operators

4.4 Fourier Integral Operators In this section, we present a brief description of basic concepts and results of the theory of Fourier integral operators. 4.4.1 Symbol Classes First, we introduce symbol classes for Fourier integral operators: Definition 4.25. Let Ω be an open subset of Rn . If m ∈ R and 0 ≤ δ < ρ ≤ 1, we let m (Ω × RN ) = the set of all functions a(x, θ) ∈ C ∞ (Ω × RN ) with Sρ,δ

the property that, for any compact K ⊂ Ω and any multi-indices α, β there exists a constant CK,α,β > 0 such that we have, for all x ∈ K and θ ∈ RN ,  α β  ∂θ ∂x a(x, θ) ≤ CK,α,β (1 + |θ|)m−ρ|α|+δ|β| . If K is a compact subset of Ω and if j is a non-negative integer, we define m a seminorm pK,j,m on Sρ,δ (Ω × RN ) by the formula   α β ∂ ∂ a(x, θ) θ x m N Sρ,δ (Ω × R )  a −→ pK,j,m (a) = sup . m−ρ|α|+δ|β| x∈K, θ∈RN (1 + |θ|) |α|≤j

m (Ω × RN ) with the topology defined by the family We equip the space Sρ,δ {pK,j,m } of seminorms where K ranges over all compact subsets of Ω and m j = 0, 1, . . .. The space Sρ,δ (Ω × RN ) is a Fr´echet space. m The elements of Sρ,δ (Ω × RN ) are called symbols of order m. We drop the m when the context is clear. Ω × RN and use Sρ,δ

We give simple and typical examples of symbols: & α Example 4.5. (1) A polynomial p(x, ξ) = of order m with |α|≤m aα (x)ξ ∞ m n coefficients in C (Ω) is in S1,0 (Ω × R ). (2) If m ∈ R, the function m/2  Ω × Rn  (x, ξ) −→ 1 + |ξ|2 m is in S1,0 (Ω × Rn ). (3) A function a(x, θ) ∈ C ∞ (Ω × (RN \ {0}) is said to be positively homogeneous of degree m in θ if it satisfies the condition

a(x, tθ) = tm a(x, θ)

for all t > 0 and θ ∈ RN \ {0}.

If a(x, θ) is positively homogeneous of degree m in θ and if ϕ(θ) is a smooth function such that ϕ(θ) = 0 for |θ| ≤ 1/2 and ϕ(θ) = 1 for |θ| ≥ 1, then the m (Ω × RN ). function ϕ(θ)a(x, θ) is in S1,0

4.4 Fourier Integral Operators

We set

S −∞ (Ω × RN ) =

,

137

m Sρ,δ (Ω × RN ).

m∈R

The next theorem gives a meaning to a formal sum of symbols of decreasing order: m

Theorem 4.26. Let aj (x, θ) ∈ Sρ,δj (Ω × RN ), mj ↓ −∞, j = 0, 1, . . .. Then m0 there exists a symbol a(x, θ) ∈ Sρ,δ (Ω × RN ), unique modulo S −∞ (Ω × RN ), such that we have, for all positive integer k, a(x, θ) −

k−1 

mk aj (x, θ) ∈ Sρ,δ (Ω × RN ).

(4.16)

j=0

If formula (4.16) holds true, we write a(x, θ) ∼

∞ 

aj (x, θ).

j=0

&∞ The formal sum j=0 aj (x, θ) is called an asymptotic expansion of a(x, θ). Now we introduce the most important symbol class that naturally enters in the study of elliptic boundary value problems: m (Ω × RN ) is said to be classical if Definition 4.27. A symbol a(x, θ) ∈ S1,0 there exist smooth functions aj (x, θ), positively homogeneous of degree m − j in θ for |θ| ≥ 1, such that we have, for all positive integer k,

a(x, θ) −

k−1 

m−k aj (x, θ) ∈ S1,0 (Ω × RN ).

j=0

The homogeneous function a0 (x, θ) of degree m is called the principal part of a(x, θ). We let m (Ω × RN ) = the set of all classical symbols of order m. Scl

For example, the symbols in Example 4.5 are all classical, and they have respectively as principal part the following functions: & (1) pm (x, ξ) = |α|=m aα (x)ξ α . (2) |ξ|m . (3) a(x, θ). m A symbol a(x, θ) in Sρ,δ (Ω × RN ) is said to be elliptic of order m if, for any compact K ⊂ Ω, there exists a positive constant CK such that

|a(x, θ)| ≥ CK (1 + |θ|)m

for all x ∈ K and |θ| ≥

1 . CK

There is a simple criterion for ellipticity in the case of classical symbols:

138

4 Lp Theory of Pseudo-Differential Operators

m Theorem 4.28. Let a(x, θ) be in Scl (Ω × RN ) with principal part a0 (x, θ). Then a(x, θ) is elliptic if and only if it satisfies the condition

a0 (x, θ) = 0

for all x ∈ Ω and |θ| = 1.

4.4.2 Phase Functions Secondly, we introduce phase functions for Fourier integral operators: Definition 4.29. Let Ω be an open subset of Rn . A function    ϕ(x, θ) ∈ C ∞ Ω × RN \ {0} is called a phase function on Ω × (RN \ {0}) if it satisfies the following three conditions (a), (b) and (c): (a) ϕ(x, θ) is real-valued. (b) ϕ(x, θ) is positively homogeneous of degree one in the variable  θ. (c) The differential dϕ(x, θ) does not vanish on Ω × RN \ {0} . We give a typical example of phase functions: Example 4.6. Let U be an open subset of Rp and Ω = U × U . The function ϕ(x, y, ξ) = (x − y) · ξ is a phase function on the space Ω × (Rp \ {0}) with n = 2p and N = p. The next lemma due to Peter Lax [75] will play a fundamental role in defining oscillatory integrals.   Lemma 4.30 (Lax). If ϕ(x, θ) is a phase function on Ω × RN \ {0} , then there exists a first-order differential operator L=

N  j=1

such that

 ∂ ∂ + bk (x, θ) + c(x, θ) ∂θj ∂xk n

aj (x, θ)

k=1

  L ei ϕ = ei ϕ ,

and that its coefficients aj (x, θ), bk (x, θ), c(x, θ) enjoy the following properties: ⎧ 0 N ⎪ ⎨aj (x, θ) ∈ S1,0 (Ω × R ), −1 bk (x, θ) ∈ S1,0 (Ω × RN ), ⎪ ⎩ −1 c(x, θ) ∈ S1,0 (Ω × RN ). Furthermore, the transpose L of L has coefficients aj (x, θ), bk (x, θ), c (x, θ) in the same symbol classes as aj (x, θ), bk (x, θ), c(x, θ), respectively.

4.4 Fourier Integral Operators

139

For example, if ϕ(x, y, ξ) is a phase function as in Example 4.6 ϕ(x, y, ξ) = (x − y) · ξ

for (x, y) ∈ U × U and ξ ∈ (Rp \ {0}),

then the operator L is given by the formula  p p   1 ξj ∂ ∂ 1 − ρ(ξ) L= √ (x − y ) + j j ∂ξj |ξ|2 ∂xk −1 2 + |x − y|2 j=1 k=1 ' p  −ξj ∂ + + ρ(ξ), |ξ|2 ∂yk k=1

where ρ(ξ) is a function in C0∞ (Rp ) such that ρ(ξ) = 1 for |ξ| ≤ 1. 4.4.3 Oscillatory Integrals If Ω is an open subset of Rn , we let "     ∞ m Ω × RN = Ω × RN . Sρ,δ Sρ,δ m∈R

  If ϕ(x, θ) is a phase function on Ω × RN \ {0} , we wish to give a meaning to the integral  Iϕ (aw) = ei ϕ(x,θ)a(x, θ) u(x) dx dθ, u ∈ C0∞ (Ω), (4.17) Ω×RN

  ∞ Ω × RN . for each symbol a(x, θ) ∈ Sρ,δ By Lemma 4.30, we can replace ei ϕ in formula (4.17) by L(ei ϕ ). Then a formal integration by parts gives us that  Iϕ (au) = ei ϕ(x,θ) L (a(x, θ)w(x, y)) dx dθ. Ω×RN

However, the properties of the coefficients of the transpose L imply that L r−η r maps Sρ,δ continuously into Sρ,δ for all r ∈ R, where η = min(ρ, 1 − δ). By continuing this process, we can reduce the growth of the integrand at infinity until it becomes integrable, and give a meaning to the integral (4.17) for each ∞ (Ω × Rn ). symbol a(x, θ) ∈ Sρ,δ More precisely, we have the following theorem: Theorem 4.31. Let 0 ≤ δ < ρ ≤ 1. Then we have the following three assertions (i), (ii) and (iii): (i) The linear functional   S −∞ Ω × RN  a −→ Iϕ (au) ∈ C

140

4 Lp Theory of Pseudo-Differential Operators

  ∞ Ω × RN whose restricextends uniquely to a linear functional  on Sρ,δ   m tion to each Sρ,δ Ω × RN is continuous. Furthermore, the restriction of   m Ω × RN is expressed as the formula the linear functional  to Sρ,δ 

ei ϕ(x,θ) (L )k (a(x, θ)w(x, y)) dx dθ,

(a) = Ω×RN

where k > (m + N )/η and η = min(ρ, 1 − δ). m Ω × RN , the mapping (ii) For any fixed a(x, θ) ∈ Sρ,δ C0∞ (Ω)  u −→ Iϕ (au) = (a) ∈ C

(4.18)

is a distribution of order ≤ k for k > (m + N )/η with η = min(ρ, 1 − δ). (iii) Let χ(θ) be a function in C0∞ (RN ) such that χ(θ) = 1 in a neighborhood of θ = 0. Then we have the formula    θ i ϕ(x,θ) e χ a(x, θ)u(x) dx dθ. (a) = lim j→∞ j Ω×RN Part (iii) in Theorem 4.31 asserts that the linear functional (a) defined in part (i) is independent of the positive integer k > (m + N )/η chosen, while part (i) asserts that (a) is independent of the function χ(θ) ∈ C0∞ (RN ) used in part (iii). ∞ an oscillatory integral, but use the We call the linear functional  on Sρ,δ standard notation as in formula (4.17). The distribution (4.18) is called the Fourier integral distribution associated with the phase function ϕ(x, θ) and the amplitude a(x, θ), and will be denoted as follows:  ei ϕ(x,θ) a(x, θ) dθ.

K(x) = RN

If u is a distribution on Ω, then the singular support of u is the smallest closed subset of Ω outside of which u is smooth. The singular support of u is denoted by sing supp u. The next theorem estimates the singular support of a Fourier integral distribution.   Theorem 4.32. If ϕ(x, θ) is a phase function on Ω × RN \ {0} and if ∞ Ω × RN , then the distribution a(x, θ) is in Sρ,δ  K(x) = RN

ei ϕ(x,θ)a(x, θ) dθ ∈ D (Ω)

satisfies the condition   sing supp K ⊂ x ∈ Ω : dθ ϕ(xθ) = 0 for some θ ∈ RN \ {0} .

4.5 Pseudo-Differential Operators

141

4.4.4 Definitions and Basic Properties of Fourier Integral Operators Let U and V be open subsets of Rp and Rq , respectively. If ϕ(x, y, θ) is a ∞ phase function on U × V × (RN \ {0}) and if a(x, y, θ) ∈ Sρ,δ (U × V × RN ),  then there is associated a distribution K ∈ D (U × V ) defined by the formula  K(x, y) = ei ϕ(x,y,θ)a(x, y, θ) dθ. RN

By applying Theorem 4.32 to our situation, we obtain that sing supp K   ⊂ (x, y) ∈ U × V : dθ ϕ(x, y, θ) = 0 for some θ ∈ RN \ {0} . The distribution K ∈ D (U × V ) defines a continuous linear operator A : C0∞ (V ) −→ D (U ) by the formula Av, u = K, u ⊗ v

for u ∈ C0∞ (U ) and v ∈ C0∞ (V ).

The operator A is called the Fourier integral operator associated with the phase function ϕ(x, y, θ) and the amplitude a(x, y, θ), and will be denoted as follows:  Av(x) = ei ϕ(x,y,θ)a(x, y, θ)v(y) dy dθ for v ∈ C0∞ (V ). V ×RN

The next theorem summarizes some basic mapping properties of the Fourier integral operator A (see Figure 4.13 below):   Theorem 4.33. (i) If dy,θ ϕ(x, y, θ) = 0 on U × V × RN \ {0} , then the operator A maps C0∞ (V ) continuously into C ∞ (U ).  (ii) If dx,θ ϕ(x, y, θ) = 0 on U × V × RN \ {0} , then the operator A extends to a continuous linear operator on E  (V ) into D (U ).   (iii) If dy,θ ϕ(x, y, θ) = 0 and dx,θ ϕ(x, y, θ) = 0 on U × V × RN \ {0} , then we have, for all v ∈ E  (V ), sing supp Av ⊂   x ∈ U : dθ ϕ(x, y, θ) = 0 for some y ∈ sing supp v and θ ∈ RN \ {0} .

4.5 Pseudo-Differential Operators In this section, we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators.

142

4 Lp Theory of Pseudo-Differential Operators A

E (V ) − −−−−→ D (U )

−−−−→ C ∞ (U ) C0∞ (V ) − A

Fig. 4.13. The mapping properties of the Fourier integral operator A

4.5.1 Definitions and Basic Properties of Pseudo-Differential Operators Let Ω1 and Ω2 be open subsets of Rn1 and Rn2 , respectively. If K(x1 , x2 ) is a distribution in D (Ω1 × Ω2 ), we can define a continuous linear operator A ∈ L (C0∞ (Ω2 ), D (Ω1 )) by the formula Aψ, ϕ = K, ϕ ⊗ ψ

for all ϕ ∈ C0∞ (Ω1 ) and ψ ∈ C0∞ (Ω2 ).

We then write A = Op (K). Since the tensor space C0∞ (Ω1 ) ⊗ C0∞ (Ω2 ) is sequentially dense in C0∞ (Ω1 × Ω2 ), it follows that the mapping D (Ω1 × Ω2 )  K −→ Op (K) ∈ L(C0∞ (Ω2 ), D (Ω1 )) is injective. The next theorem asserts that it is also surjective ([26, Chapitre I, Th´eor`eme 4.4], [122, Theorem 5.36]): Theorem 4.34 (the Schwartz kernel theorem). If A is a continuous linear operator on C0∞ (Ω2 ) into D (Ω1 ), then there exists a unique distribution KA (x1 , x2 ) in D (Ω1 × Ω2 ) such that A = Op (K). The distribution KA is called the kernel of A. Formally, we have the formula  KA (x1 , x2 ) ψ(x2 ) dx2 for all ψ ∈ C0∞ (Ω2 ). Aψ(x1 ) = Ω2

Now we give some important examples of distributions kernels (see Table 4.2 below and also Example 4.2): Example 4.7. (a) Riesz potentials: Ω1 = Ω2 = Rn and 0 < α < n. −α/2

u(x) = Rα ∗ u(x)  Γ ((n − α)/2) 1 = α n/2 u(y) dy 2 π Γ (α/2) Rn |x − y|n−α (−Δ)

for u ∈ C0∞ (Rn ).

4.5 Pseudo-Differential Operators

143

(b) Newtonian potentials: Ω1 = Ω2 = Rn for n ≥ 3. −1

(−Δ)

u(x) = N ∗ u(x)  1 u(y) dy n−2 Rn |x − y|

Γ ((n − 2)/2) = 4 π n/2

for u ∈ C0∞ (Rn ).

(c) Bessel potentials: Ω1 = Ω2 = Rn and α > 0. (I − Δ)

−α/2

= Gα ∗ u(x) =

u(x)  Rn

Gα (x − y) u(y) dy

for u ∈ C0∞ (Rn ).

(d) Riesz operators: Ω1 = Ω2 = Rn and 1 ≤ j ≤ n. Yj u(x) = Rj ∗ u(x)  √ Γ ((n + 1)/2) xj − yj = −1 v. p. u(y) dy |x − y|n+1 π (n+1)/2 n R

for u ∈ C0∞ (Rn ).

(e) The Calder´ on–Zygmund integro-differential operator: Ω1 = Ω2 = Rn .   n ∂u 1  (−Δ)1/2 u(x) = √ Yj (x) ∂xj −1 j=1  n xj − yj ∂u Γ ((n + 1)/2)  v. p. (y) dy = n+1 ∂y (n+1)/2 π j Rn |x − y| j=1

for u ∈ C0∞ (Rn ).

Now we are in a position to define pseudo-differential operators: Definition 4.35. Let Ω be an open subset of Rn and m ∈ R. A pseudodifferential operator of order m on Ω is a Fourier integral operator of the form  ei(x−y)·ξ a(x, y, ξ)u(y) dydξ for u ∈ C0∞ (Ω), (4.19) Au(x) = Ω×Rn

m with some a(x, y, ξ) ∈ Sρ,δ (Ω × Ω × Rn ). In other words, a pseudo-differential operator of order m is a Fourier integral operator associated with the phase m (Ω × Ω × function ϕ(x, y, ξ) = (x − y) · ξ and some amplitude a(x, y, ξ) ∈ Sρ,δ n R ).

We let Lm ρ,δ (Ω) = the set of all pseudo-differential operators of order m on Ω. The set Lm ormander class. ρ,δ (Ω) is called the H¨ By applying Theorems 4.32 and 4.33 to our situation, we obtain the following three assertions (1), (2) and (3):

144

4 Lp Theory of Pseudo-Differential Operators

Operators

Notation

Distribution kernels

Riesz potential

(−Δ)−α/2 (0 < α < n)

Γ ((n−α)/2) 1 2α π n/2 Γ (α/2) |x|n−α

Newtonian potential

(−Δ)−1 (α = 2)

Γ ((n−2)/2) 1 |x|n−2 4π n/2

Bessel potential

(I − Δ)−α/2 (α > 0)

Riesz operator

Yj (1 ≤ j ≤ n)

Gα (x)

i

Γ ((n+1)/2) π (n+1)/2

x

j v. p. |x|n+1

Table 4.2. Operators and their kernels in Example 4.7 A

E (Ω) − −−−− → D (Ω)

−−−− → C ∞ (Ω) C0∞ (Ω) − A

Fig. 4.14. The mapping properties of the pseudo-differential operator A

(1) A pseudo-differential operator A maps C0∞ (Ω) continuously into C ∞ (Ω) and extends to a continuous linear operator A : E  (Ω) → D (Ω) (see Figure 4.14 above). (2) The distribution kernel KA (x, y), defined by the formula KA (x, y) =

1 (2π)n

 ei(x−y)·ξ a(x, y, ξ) dξ, Rn

of a pseudo-differential operator A satisfies the condition sing supp KA ⊂ {(x, x) : x ∈ Ω} , that is, the kernel KA is smooth off the diagonal {(x, x) : x ∈ Ω} in Ω ×Ω.

4.5 Pseudo-Differential Operators

145

(3) sing supp Au ⊂ sing supp u, u ∈ E  (Ω).In other words, Au is smooth whenever u is. This property is referred to as the pseudo-local property. We set

L−∞ (Ω) =

,

Lm ρ,δ (Ω).

m∈R

The next theorem characterizes the class L−∞ (Ω). Theorem 4.36. The following three conditions (i), (ii) and (iii) are equivalent: (i) A ∈ L−∞ (Ω). (ii) A is written in the form (4.19) with some a ∈ S −∞ (Ω × Ω × Rn ). (iii) A is a regularizer, or equivalently, its distribution kernel KA (x, y) is in the space C ∞ (Ω × Ω). The next definition plays an important role in the study of partial differential operators: Definition 4.37. A continuous linear operator A : C0∞ (Ω) −→ D (Ω) is said to be properly supported if the following two conditions (a) and (b) are satisfied (see Figures 4.15 and 4.16 below): (a) For any compact subset K of Ω, there exists a compact subset K  of Ω such that supp v ⊂ K =⇒ supp Av ⊂ K  . (b) For any compact subset K  of Ω, there exists a compact subset K ⊃ K  of Ω such that supp v ∩ K = ∅ =⇒ supp Av ∩ K  = ∅. If A is properly supported, then it maps C ∞ (Ω) continuously into itself, and extends to a continuous linear operator on D (Ω) into itself. The situation can be visualized as in Figure 4.17 below. The next theorem states that every pseudo-differential operator can be written as the sum of a properly supported operator and a regularizer. Theorem 4.38. If A ∈ Lm ρ,δ (Ω), then we have the decomposition A = A0 + R, −∞ where A0 ∈ Lm (Ω). ρ,δ (Ω) is properly supported and R ∈ L

Proof. Take a function σ ∈ C ∞ (Ω × Ω) (see [41, Proposition (8.15)]) that satisfies the following two conditions (a) and (b):

146

4 Lp Theory of Pseudo-Differential Operators supp A

Ω

K

Ω K

Fig. 4.15. Condition (a) in Definition 4.37 supp A

Ω

K

Ω K

Fig. 4.16. Condition (b) in Definition 4.37 A

D (Ω) − −−−− → D (Ω)

−−−− → C ∞ (Ω) C ∞ (Ω) − A

Fig. 4.17. The mapping properties of the properly supported, pseudo-differential operator A

(a) σ(x, y) = 1 in a neighborhood of the diagonal ΔΩ = {(x, x) : x ∈ Ω} in Ω × Ω; (b) The restrictions to supp σ of the projections pi : Ω × Ω  (x1 , x2 ) −→ xi ∈ Ω are proper mappings.

for i = 1, 2

4.5 Pseudo-Differential Operators

147

Then it is easy to see that the operators A0 and R, defined respectively by the distribution kernels KA0 (x, y) = σ(x, y) KA (x, y), KR (x, y) = (1 − σ(x, y)) KA (x, y), are the desired ones, since the distribution kernel KA is of class C ∞ off the diagonal ΔΩ . The proof of Theorem 4.38 is complete.   m (Ω×Rn ), then the operator p(x, D), defined by the formula If p(x, ξ) ∈ Sρ,δ

p(x, D)u(x) =

1 (2π)n

 Rn

ei x·ξ p(x, ξ) u (ξ) dξ

for u ∈ C0∞ (Ω),

(4.20)

is a pseudo-differential operator of order m on Ω, that is, p(x, D) ∈ Lm ρ,δ (Ω). The next theorem asserts that every properly supported pseudo-differential operator can be reduced to the form (4.20). Theorem 4.39. If A ∈ Lm ρ,δ (Ω) is properly supported, then we have the assertion m (Ω × Rn ), p(x, ξ) = e−i x·ξ A(ei x·ξ ) ∈ Sρ,δ and A = p(x, D). × Ω × Rn ) is an amplitude for A, then we Furthermore, if a(x, y, ξ) ∈ have the following asymptotic expansion:   1  α α ∂ D (a(x, y, ξ)) p(x, ξ) ∼ . α! ξ y y=x m (Ω Sρ,δ

α≥0

The function p(x, ξ) is called the complete symbol of A. We can extend the notion of a complete symbol to the whole space Lm ρ,δ (Ω) in the following way: If A ∈ Lm (Ω), then we choose a properly supported ρ,δ −∞ operator A0 ∈ Lm (Ω), and define ρ,δ (Ω) such that A − A0 ∈ L σ(A) = the equivalence class of the complete symbol of A0 m in Sρ,δ (Ω × Rn )/S −∞ (Ω × Rn ).

In view of Theorems 4.38 and 4.39, it follows that σ(A) does not depend on the operator A0 chosen. The equivalence class σ(A) is called the complete symbol of A. It is easy to see that the mapping m n −∞ (Ω × Rn ) Lm ρ,δ (Ω)  A −→ σ(A) ∈ Sρ,δ (Ω × R )/S

induces an isomorphism

148

4 Lp Theory of Pseudo-Differential Operators −∞ m Lm (Ω) −→ Sρ,δ (Ω × Rn )/S −∞ (Ω × Rn ). ρ,δ (Ω)/L

We shall often identify the complete symbol σ(A) with a representative m in the class Sρ,δ (Ω × Rn ) for notational convenience, and call any member of σ(A) a complete symbol of A. Now we intro duce the most important H¨ ormander class that enter naturally in connection with the classical symbol class: Definition 4.40. A pseudo-differential operator A ∈ Lm 1,0 (Ω) is said to be m classical if its complete symbol σ(A) has a representative in the class Scl (Ω × n R ). We let Lm cl (Ω) = the set of all classical pseudo-differential operators of order m on Ω. Then the mapping m n −∞ Lm (Ω × Rn ) cl (Ω)  A −→ σ(A) ∈ Scl (Ω × R )/S

induces an isomorphism −∞ m Lm (Ω) −→ Scl (Ω × Rn )/S −∞ (Ω × Rn ). cl (Ω)/L

Also we have the assertion L−∞ (Ω) =

,

Lm cl (Ω).

m∈R

If A ∈ Lm cl (Ω), then the principal part of σ(A) has a canonical representative σA (x, ξ) ∈ C ∞ (Ω × (Rn \ {0})) which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. The next two theorems assert that the class of pseudo-differential operators forms an algebra closed under the operations of composition of operators and taking the transpose or adjoint of an operator.  ∗ Theorem 4.41. If A ∈ Lm ρ,δ (Ω), then its transpose A and its adjoint A are m  ∗ both in Lρ,δ (Ω), and the complete symbols σ(A ) and σ(A ) have respectively the following asymptotic expansions:

 1 ∂ α Dα (σ(A)(x, −ξ)) , α! ξ x α≥0    1 σ(A∗ )(x, ξ) ∼ ∂ξα Dxα σ(A)(x, ξ) . α! σ(A )(x, ξ) ∼

α≥0

4.5 Pseudo-Differential Operators 

149



m   Theorem 4.42. If A ∈ Lm ρ ,δ  (Ω) and B ∈ Lρ ,δ  (Ω) where 0 ≤ δ < ρ ≤ 1 and if one of them is properly supported, then the composition AB is in  +m (Ω) with ρ = min(ρ , ρ ), δ = max(δ  , δ  ), and we have the following Lm ρ,δ asymptotic expansion:

σ(AB)(x, ξ) ∼

 1 ∂ α (σ(A)(x, ξ)) · Dxα (σ(B)(x, ξ)) . α! ξ

α≥0

We introduce the special H¨ormander class that are the “invertible” elements in the algebra of pseudo-differential operators: Definition 4.43. A pseudo-differential operator A ∈ Lm ρ,δ (Ω) is said to be elliptic of order m if its complete symbol σ(A) is elliptic of order m. In view of Theorem 4.28, it follows that a classical pseudo-differential operator A ∈ Lm cl (Ω) is elliptic if and only if its homogeneous principal symbol σA (x, ξ) does not vanish on the space Ω × (Rn \ {0}). In fact, we can prove the following theorem: Theorem 4.44. An operator A ∈ Lm ρ,δ (Ω) is elliptic if and only if there exists a properly supported operator B ∈ L−m ρ,δ (Ω) such that: 

AB ≡ I BA ≡ I

mod L−∞ (Ω), mod L−∞ (Ω).

Such an operator B is called a parametrix for A. In other words, a parametrix for A is a two-sided inverse of A modulo L−∞ (Ω). We observe that a parametrix is unique modulo L−∞ (Ω). The next theorem proves the invariance of pseudo-differential operators under change of coordinates. Theorem 4.45. Let Ω1 and Ω2 be two open subsets of Rn and let χ : Ω1 → Ω2 be a C ∞ diffeomorphism. If A ∈ Lm ρ,δ (Ω1 ), where 1 − ρ ≤ δ < ρ ≤ 1, then the mapping Aχ : C0∞ (Ω2 ) −→ C ∞ (Ω2 ) v −→ A(v ◦ χ) ◦ χ−1 is in Lm ρ,δ (Ω2 ), and we have the asymptotic expansion σ(Aχ )(y, η) ∼

   1    ∂ξα σ(A) (x, tχ (x) · η) · Dzα ei r(x,z,η)  α! z=x

(4.21)

α≥0

with

r(x, z, η) = χ(z) − χ(x) − χ (x) · (z − x), η .

Here x = χ−1 (y),χ (x) is the derivative of χ at x and tχ (x) is its transpose.

150

4 Lp Theory of Pseudo-Differential Operators

Remark 4.46. Formula (4.21) shows that   σ(Aχ )(y, η) ≡ σ(A) x, tχ (x) · η

m−(ρ−δ)

mod Sρ,δ

.

Note that the mapping   Ω2 × Rn  (y, η) −→ x, tχ (x) · η ∈ Ω1 × Rn is just a transition map of the cotangent bundle T ∗ (Rn ). This implies that n the principal symbol σm (A) of A ∈ Lm ρ,δ (R ) can be invariantly defined on ∗ n T (R ) when 1 − ρ ≤ δ < ρ ≤ 1. The situation may be visualized as in Figure 4.18 below. A

C0∞ (Ω1 ) − −−−− → C ∞ (Ω1 ) χ∗

χ∗

C0∞ (Ω2 ) − −−−− → C ∞ (Ω2 ) Aχ

Fig. 4.18. The pseudo-differential operators A and Aχ in Theorem 4.45

Here χ∗ v = v ◦ χ is the pull-back of v by χ and χ∗ u = u ◦ χ−1 is the pushforward of u by χ, respectively. 4.5.2 Lp Boundedness of Pseudo-Differential Operators In this subsection, we study the boundedness of pseudo-differential operators in the framework of Lp Sobolev spaces. A differential operator of order m with smooth coefficients on Ω is cons,p s,p s−m,p s−m,p (Ω) (resp. Bloc (Ω)) into Hloc (Ω) (resp. Bloc (Ω)) for all tinuous on Hloc s ∈ R. This result extends to pseudo-differential operators: More precisely, we can obtain the following Lp boundedness theorem for properly supported, pseudo-differential operators due to Bourdaud [16]: Theorem 4.47 (the Besov space boundedness theorem). Every properly supported operator A ∈ Lm 1,δ (Ω) with 0 ≤ δ < 1 extends to continuous linear operators (see Figures 4.19 and 4.20 below) s,p s−m,p A : Hloc (Ω) −→ Hloc (Ω)

A:

s,p (Ω) Bloc

−→

s−m,p Bloc (Ω)

for all s ∈ R and 1 < p < ∞, for all s ∈ R and 1 ≤ p ≤ ∞.

For a proof of Theorem 4.47, the reader might refer to Bourdaud [16, Theorem 1] (see also [122, Appendix A]).

4.5 Pseudo-Differential Operators A

D (Ω) − −−−−→

151

D (Ω)

A

s,p s−m,p (Ω) − −−−−→ Hloc (Ω) Hloc

−−−−→ C ∞ (Ω) − A

C ∞ (Ω)

s,p Fig. 4.19. The mapping properties of A in the localized Sobolev spaces Hloc (Ω)

A

D (Ω) − −−−−→

D (Ω)

A

s,p s−m,p (Ω) − −−−−→ Bloc (Ω) Bloc

−−−−→ C ∞ (Ω) − A

C ∞ (Ω)

s,p Fig. 4.20. The mapping properties of A in the localized Besov spaces Bloc (Ω)

4.5.3 Pseudo-Differential Operators on a Manifold Now we define the concept of a pseudo-differential operator on a manifold, and transfer all the machinery of pseudo-differential operators to manifolds. Let M be an n-dimensional compact smooth manifold without boundary. Theorem 4.45 leads us to the following definition (see Figure 4.21 below): Definition 4.48. Let 1 − ρ ≤ δ < ρ ≤ 1. A continuous linear operator A : C ∞ (M ) → C ∞ (M ) is called a pseudo-differential operator of order m ∈ R if it satisfies the following two conditions (i) and (ii): (i) The distribution kernel KA (x, y) of A is smooth off the diagonal {(x, x) : x ∈ M } in M × M . (ii) For any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) −→ C ∞ (χ(U )) v −→ A (v ◦ χ) ◦ χ−1 belongs to the class Lm ρ,δ (χ(U )).

4 Lp Theory of Pseudo-Differential Operators

152

C0∞ (U )

A|U

− −−−− →

χ∗

C ∞ (U ) χ∗

C0∞ (χ(U )) − −−−− → C ∞ (χ(U )) Aχ

Fig. 4.21. The mapping Aχ = χ∗ (A|U ) χ∗ in Definition 4.48

We let Lm ρ,δ (M ) = the set of all pseudo-differential operators of order m on M , and set

L−∞ (M ) =

,

Lm ρ,δ (M ).

m∈R

Some results about pseudo-differential operators on Rn stated above are also true for pseudo-differential operators on M . In fact, pseudo-differential operators on M are defined to be locally pseudo-differential operators on Rn . For example, we have the following five assertions (1) through (5) (see Figures 4.22 and 4.23 below): (1) A pseudo-differential operator A extends to a continuous linear operator A : D (M ) → D (M ). (2) sing supp Au ⊂ sing supp u, u ∈ D (M ). (3) A continuous linear operator A : C ∞ (M ) → D (M ) is a —em regularizer if and only if it is in the class L−∞ (M ). (4) The class Lm ρ,δ (M ) is stable under the operations of composition of operators and taking the transpose or adjoint of an operator. (5) (The Besov space boundedness theorem) A pseudo-differential operator A ∈ Lm 1,δ (M ) with 0 ≤ δ < 1 extends to continuous linear operators (see Figures 4.22 and 4.23) A : H s,p (M ) −→ H s−m,p (M ) for all s ∈ R and 1 < p < ∞, A : B s,p (M ) −→ B s−m,p (M ) for all s ∈ R and 1 ≤ p ≤ ∞. A pseudo-differential operator A ∈ Lm 1,0 (M ) is said to be classical if, for any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) → C ∞ (χ(U )) belongs to the class Lm cl (χ(U )). We let Lm cl (M ) = the set of all classical, pseudo-differential operators of order m on M .

4.5 Pseudo-Differential Operators A

D (M ) − −−−−→

153

D (M )

A

−−−−→ H s−m,p (M ) H s,p (M ) −

−−−−→ C ∞ (M ) − A

C ∞ (M )

Fig. 4.22. The mapping properties of A in the Sobolev spaces H s,p (M ) A

D (M ) − −−−−→

D (M )

A

−−−−→ B s−m,p (M ) B s,p (M ) −

−−−−→ C ∞ (M ) − A

C ∞ (M )

Fig. 4.23. The mapping properties of A in the Besov spaces B s,p (M )

We observe that

L−∞ (M ) =

,

Lm cl (M ).

m∈R

Let A ∈ Lm cl (M ). If (U, χ) is a chart on M , there is associated a homogeneous principal symbol σAχ ∈ C ∞ (χ(U ) × (Rn \ {0})). In view of Remark 4.46, by smoothly patching together the functions σAχ we can obtain a smooth function σA (x, ξ) on T ∗ (M ) \ {0} = {(x, ξ) ∈ T ∗ (M ) : ξ = 0}, which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. A classical pseudo-differential operator A ∈ Lm cl (M ) is said to be elliptic of order m if its homogeneous principal symbol σA (x, ξ) does not vanish on the bundle T ∗ (M ) \ {0} of non-zero cotangent vectors. Then we have the following assertion (6): (6) An operator A ∈ Lm cl (M ) is elliptic if and only if there exists a parametrix (M ) for A: B ∈ L−m cl  AB ≡ I mod L−∞ (M ), BA ≡ I mod L−∞ (M ).

154

4 Lp Theory of Pseudo-Differential Operators

4.5.4 Hypoelliptic Pseudo-Differential Operators Let Ω be an open subset of Rn . A properly supported pseudo-differential operator A on Ω is said to be hypoelliptic if it satisfies the condition sing supp u = sing supp Au

for all u ∈ D (Ω).

For example, Theorem 4.44 asserts that elliptic operators are hypoelliptic. We remark that this notion may be transferred to manifolds. The following criterion for hypoellipticity is due to H¨ormander (cf. [59, Theorem 4.2]): Theorem 4.49 (H¨ ormander). Let A = p(x, D) be a properly supported pseudo-differential operator in the H¨ ormander class Lm ρ,δ (Ω) with 1 − ρ ≤ δ < ρ ≤ 1. Assume that, for any compact K ⊂ Ω and any multi-indices α, β there exist constants CK,α,β > 0, CK > 0 and μ ∈ R such that we have, for all x ∈ K and |ξ| ≥ CK ,   α β Dξ Dx p(x, ξ) ≤ CK,α,β |p(x, ξ)| (1 + |ξ|)−ρ|α|+δ|β| , −1

|p(x, ξ)|

μ

≤ CK (1 + |ξ|) .

(4.22a) (4.22b)

Then there exists a parametrix B ∈ Lμρ,δ (Ω) for A. Remark 4.50. It should be emphasized that Theorem 4.49 extends to the n H¨ormander class Lm ρ,δ (Ω, R ) of n × n matrix-valued, pseudo-differential operators on Ω (see [61, Chapter XXII, Theorem 22.1.3]; [119]). 4.5.5 Distribution Kernel of Pseudo-Differential Operators In this subsection, following Coifman–Meyer [30, Chapitre IV, Proposition 1]) we state that the distribution kernel  1 s(x, y) = ei(x−y)·ξ a(x, ξ) dξ (2π)n Rn n of a pseudo-differential operator S ∈ Lm 1,0 (R ) with symbol a(x, θ) satisfies the estimate

|s(x, y)| ≤

C |x − y|n+m

for all x, y ∈ Rn and x = y.

More precisely, we can obtain the following fundamental result (see [122, Section 7.8, Theorem 7.36]): m Theorem 4.51. Let σ(x, ξ) be a symbol in the class S1,0 (Rn × Rn ) such that    (α)  m−|α| for all (x, ξ) ∈ Rn × Rn . (4.23) σ(β) (x, ξ) ≤ Cαβ (1 + |ξ|)

4.5 Pseudo-Differential Operators

We let

1 r(x, z) = (2π)n

155

 ei z·ξ σ(x, ξ) dξ, Rn

where the integral is taken in the sense of oscillatory integrals (see Theorem 4.31). Then the distribution r(x, z) satisfies the condition r(x, z) ∈ C ∞ (Rn × Rn \ {0}) , and the estimate |r(x, z)| ≤

C n+m

|z|

for all z = 0.

(4.24)

We can summarize the three operators (−Δ)−α/2 , (−Δ)−1 and (I −Δ)−α/2 in Example 4.7 as in Table 4.3 below.

Notation

Distribution kernels

Symbols

(−Δ)−α/2 (0 < α < n)

Γ ((n−α)/2) 1 2α π n/2 Γ (α/2) |x|n−α

1 |ξ|α

(−Δ)−1 (α = 2)

Γ ((n−2)/2) 1 |x|n−2 4π n/2

1 |ξ|2

Gα (x)

1 (1+|ξ|2 )α/2

(I − Δ)−α/2 (α > 0)

Table 4.3. Riesz, Newtonian and Bessel potentials in Example 4.7

For the calculation of their distribution kernels via the heat kernel, the reader might refer to Section 4.7. Moreover, we give an example of the density function of L´evy measure in the symmetric α-stable process for 0 < α < 2: α/2

n Example 4.8. Let 0 < α < 2. If Aα = − (−Δ) ∈ Lα 1,0 (R ) is a pseudoα differential operator with symbol −|ξ| , then we find from Example 4.3 that its distribution kernel aα (x, y) is equal to the following:

aα (x, y) =

1 2α Γ ((α + n)/2) v.p. . |x − y|n+α π n/2 |Γ (−α/2)|

4 Lp Theory of Pseudo-Differential Operators

156

Hence we have, for every f ∈ C0∞ (Rn ),

 1 Aα f (x) = − (−Δ) f (x) = ei x·ξ (−|ξ|α ) f(ξ) dξ (2π)n Rn  2α Γ ((α + n)/2) 1 = n/2 v.p. f (y) dy |x − y|n+α π |Γ (−α/2)| Rn  f (x + y) − f (x) 2α Γ ((α + n)/2) = n/2 dy lim |y|n+α π |Γ (−α/2)| ε↓0 |y|≥ε α/2

for 0 < α < 1, and also

 1 f (x) = ei x·ξ (−|ξ|α ) f(ξ) dξ (2π)n Rn  2α Γ ((α + n)/2) 1 = n/2 v.p. f (y) dy |x − y|n+α π |Γ (−α/2)| Rn  f (x + y) + f (x − y) − 2f (x) 2α Γ ((α + n)/2) lim = n/2 dy ε↓0 |y|n+α π |Γ (−α/2)| |y|≥ε α/2

Aα f (x) = − (−Δ)

for 1 ≤ α < 2. We remark (see Table 4.4 below) that the distribution να (y) :=

2α Γ ((α + n)/2) 1 v.p. n+α |y| π n/2 |Γ (−α/2)|

is the density function of L´evy measure in the symmetric α-stable process (see [63, Chapter 1]) and further that the operator Aα is the infinitesimal generator of the probabilistic convolution semigroup {et Aα }t≥0 (see [27, Theorem 1.4]). More precisely, we have the formula  t Aα α f (x) = (Pt ∗ f ) (x) = Ptα (x − y)f (y) dy for all f ∈ C0∞ (Rn ), e Rn

where Ptα (x)

1 = (2π)n

 Rn

α

ei x·ξ e−t |ξ| dξ

for x ∈ Rn and t > 0.

4.6 Elliptic Pseudo-differential Operators and their Indices In this section, by using the Riesz–Schauder theory we prove some of the most important results about elliptic pseudo-differential operators on a manifold and their indices in the framework of Sobolev spaces. These results will be useful for the study of elliptic boundary value problems in Chapter 5. Throughout this section, let M be an n-dimensional, compact smooth manifold without boundary.

4.6 Elliptic Pseudo-differential Operators and their Indices

Aα = − (−Δ)α/2

Infinitesimal generator

Distribution kernel

157

2α Γ ((α+n)/2) π n/2 |Γ (−α/2)|

v.p.

1 |x − y|n+α

−|ξ|α

Symbol

Table 4.4. An overview of symmetric α-stable processes for 0 < α < 2

4.6.1 Pseudo-Differential Operators on Sobolev Spaces Let H s (M ) = H s,2 (M ) (p = 2) be the Sobolev space of order s ∈ R on M . Recall that , H s (M ), C ∞ (M ) = s∈R



D (M ) =

"

H s (M ).

s∈R

A linear operator T : C ∞ (M ) → C ∞ (M ) is said to be of order m ∈ R if it extends to a continuous linear operator on H s (M ) into H s−m (M ) for each s ∈ R. For example, every pseudo-differential operator in Lm (M ) is of order m. We say that T : C ∞ (M ) → C ∞ (M ) is of order −∞ if it extends to a continuous linear operator on H s (M ) into C ∞ (M ) for each s ∈ R. This is equivalent to saying that T is a regularizer; hence we have the formula L−∞ (M ) = the set of all operators of order −∞.

(4.25)

Let T : H s (M ) → H t (M ) be a linear operator with domain D(T ) dense in H (M ). Each element v of H −t (M ) defines a linear functional G on D(T ) by the formula G(u) = (T u, v) for u ∈ D(T ), s

where (·, ·) on the right-hand side is the sesquilinear pairing of H t (M ) and H −t (M ). If this functional G is continuous everywhere on D(T ), by applying [123, Theorem 2.6] we obtain that G can be extended uniquely to a continuous 0 on D(T ) = H s (M ). Hence, there exists a unique element linear functional G ∗ −s v of H (M ) such that

158

4 Lp Theory of Pseudo-Differential Operators

0 G(u) = (u, v ∗ )

for u ∈ H s (M ),

since the sesquilinear form (·, ·) on the product space H s (M ) × H −s (M ) permits us to identify the strong dual space of H s (M ) with H −s (M ). In particular, we have the formula (T u, v) = G(u) = (u, v ∗ )

for all u ∈ D(T ).

So we let D(T ∗ ) = the totality of those v ∈ H −t (M ) such that the mapping u → (T u, v) is continuous everywhere on D(T ), and define

T ∗v = v∗ .

Therefore, it follows that T ∗ is a linear operator from H −t (M ) into H −s (M ) with domain D (T ∗ ) such that (T u, v) = (u, T ∗ v)

for all u ∈ D(T ) and v ∈ D (T ∗ ).

(4.26)

The operator T ∗ is called the adjoint of T . The transpose of T is a linear operator T  from H −t (M ) into H −s (M ) with domain D(T  ) such that D (T  ) = the totality of those v ∈ H −t (M ) such that the mapping u → T u, v is continuous everywhere on D(T ), and satisfies the formula T u, v = u, T  v

for all u ∈ D(T ) and v ∈ D (T  ).

(4.27)

Here ·, · on the left-hand (resp. right-hand) side is the bilinear pairing of H t (M ) and H −t (M ) (resp. H s (M ) and H −s (M )). In view of formulas (4.26) and (4.27), it follows that (a) v ∈ D(T  ) ⇐⇒ v ∈ D (T ∗ ), (b) T  v = T ∗ v for every v ∈ D(T  ). Here · denotes complex conjugation. Hence, we have the following two assertions (1) and (2):  (1) The ranges R (T ∗ ) and R (T  ) are isomorphic. (2) The null spaces N (T ∗ ) and N (T  ) are isomorphic.

(4.28)

Now let A ∈ Lm (M ). Then the operator A : C ∞ (M ) → C ∞ (M ) extends uniquely to a continuous linear operator As : H s (M ) −→ H s−m (M )

4.6 Elliptic Pseudo-differential Operators and their Indices A

D (M ) − −−−− →

159

D (M )

A

H s (M ) − −−−s− → H s−m (M )

−−−− → C ∞ (M ) − A

C ∞ (M )

Fig. 4.24. The mapping properties of the continuous operators A, As and A

for all s ∈ R, and hence to a continuous linear operator A : D (M ) −→ D (M ). The situation can be visualized as in Figure 4.24 above. The adjoint A∗ of A is also in Lm (M ); hence the operator A∗ : C ∞ (M ) −→ C ∞ (M ) extends uniquely to a continuous linear operator A∗s : H s (M ) −→ H s−m (M ) for all s ∈ R. The next lemma states a fundamental relationship between the operators As and A∗s .: Lemma 4.52. If A ∈ Lm (M ), we have, for all s ∈ R,  ∗ (As ) = A∗−s+m ,  ∗ ∗ A−s+m = As .

(4.29)

Proof. If u ∈ D(As ) = H s (M ) and v ∈ D(A∗−s+m ) = H −s+m (M ), there exist sequences {uj } and {vj } in C ∞ (M ) such that  uj −→ u in H s (M ) as j → ∞, vj −→ v in H −s+m (M ) as j → ∞. Then we have the assertions  Auj −→ As u A∗ vj −→ A∗−s+m v so that

in H s−m (M ), in H −s (M ),

  (As u, v) = lim (Auj , vj ) = lim (uj , A∗ vj ) = u, A∗−s+m v . j→∞

j→∞

This proves the desired formulas (4.29). The proof of Lemma 4.52 is complete.  

160

4 Lp Theory of Pseudo-Differential Operators

4.6.2 The Index of an Elliptic Pseudo-Differential Operator In this section, we study the operators As when A is a classical elliptic pseudodifferential operator. The next theorem is an immediate consequence of Theorem 4.44: Theorem 4.53 (the elliptic regularity theorem). Let A ∈ Lm cl (M ) be elliptic. Then we have, for all s ∈ R, u ∈ D (M ), Au ∈ H s (M ) =⇒ u ∈ H s+m (M ). In particular, we have the assertions R (As ) ∩ C ∞ (M ) = R(A), N (As ) = N (A).

(4.30) (4.31)

Here R (As ) = {Au : u ∈ H s (M )} ,

R(A) = {Au : u ∈ C ∞ (M )} ;

N (As ) = {u ∈ H (M ) : As u = 0} , s

N (A) = {u ∈ C ∞ (M ) : Au = 0} .

Here it is worth pointing out that assertion (4.31) is a generalization of the celebrated Weyl theorem, which states that harmonic functions on Euclidean space are smooth. Now let X, Y be Banach spaces. A bounded (continuous) linear operator T : X −→ Y is called a Fredholm operator if it satisfies the following three conditions (i), (ii) and (iii): (i) The null space N (T ) = {x ∈ X : T x = 0} of T has finite dimension, that is, dim N (T ) < ∞. (ii) The range R(T ) = {T x : x ∈ X} of T is closed in X. (iii) The range R(T ) has finite codimension in X, that is, codim R(T ) = dim X/R(T ) < ∞. In this case the index ind T of T is defined by the formula ind T = dim N (T ) − codim R(T ). The next theorem states that the operator As : H s (M ) → H s−m (M ) is a Fredholm operator for every s ∈ R: s Theorem 4.54. If A ∈ Lm cl (M ) is elliptic, the operator As : H (M ) → s−m H (M ) is a Fredholm operator for every s ∈ R.

4.6 Elliptic Pseudo-differential Operators and their Indices

161

Proof. Take a parametrix B ∈ L−m cl (M ) for A:  BA = I + P, P ∈ L−∞ (M ), AB = I + Q, Q ∈ L−∞ (M ). Then we have, for all s ∈ R,  Bs−m · As = I + Ps , As · Bs−m = I + Qs−m . Furthermore, in view of assertion (4.25) it follows from an application of the Rellich–Kondrachov theorem (Theorem 4.10) that the operators Ps : H s (M ) −→ H s (M ) and Qs−m : H s−m (M ) −→ H s−m (M ) are both compact. Therefore, by applying [100, Theorem 7.2] (cf. [80, Theorem 2.24]) to our situation we obtain that As is a Fredholm operator. The proof of Theorem 4.54 is complete.   Corollary 4.55. Let A ∈ Lm cl (M ) be elliptic. Then we have the following two assertions (i) and (ii): (i) The range R(A) of A is a closed linear subspace of C ∞ (M ). (ii) The null space N (A) of A is a finite dimensional, closed linear subspace of C ∞ (M ). Proof. (i) It follows from Theorem 4.54 that the range R (As ) of As is closed in H s−m (M ); hence it is closed in C ∞ (M ), since the injection C ∞ (M ) −→ H s−m (M ) is continuous. In view of formula (4.30), this proves part (i). (ii) Similarly, in view of formula (4.31) it follows from Theorem 4.54 that N (A) has finite dimension; so it is closed in each H s (M ) and hence in C ∞ (M ). The proof of Corollary 4.55 is complete.   The next theorem asserts that the index ind As = dim N (As ) − codim R(As ) is independent of s ∈ R: Theorem 4.56. If A ∈ Lm cl (M ) is elliptic, then we have, for all s ∈ R, ind As = dim N (A) − dim N (A∗ ), where

N (A∗ ) = {v ∈ C ∞ (M ) : A∗ v = 0} .

(4.32)

162

4 Lp Theory of Pseudo-Differential Operators

Proof. Since the range R (As ) is closed in H s−m (M ), by applying the closed range theorem (see [100, Theorem 3.16], [147, p. 205, Theorem]) to our situation we obtain that codim R (As ) = dim N (As ) . However, in view of assertion (4.28) it follows that dim N (As ) = dim N (A∗s ) . Furthermore, we have, by formulas (4.29) and (4.31), N (A∗s ) = N (A∗−s+m ) = N (A∗ ),

(4.33)

since A∗ ∈ Lm cl (M ) is also elliptic (cf. Theorem 4.41). Summing up, we obtain that codim R(As ) = dim N (A∗ ).

(4.34)

Therefore, the desired formula (4.32) follows from formulas (4.31) and (4.34). The proof of Theorem 4.56 is complete.   We give another useful expression for ind As . To do this, we need the following lemma: ∗ Lemma 4.57. Let A ∈ Lm cl (M ) be elliptic. Then the spaces N (A ) and R(A) ∞ are orthogonal complements of each other in the space C (M ) relative to the inner product of L2 (M ) (see Figure 4.25 below):

C ∞ (M ) = N (A∗ ) ⊕ R(A).

C ∞ (M )

N (A∗ ) = N (A∗m )

0 R(A) = R(Am ) ∩ C ∞ (M )

Fig. 4.25. The orthogonal decomposition (4.35) in the space C ∞ (M )

(4.35)

4.6 Elliptic Pseudo-differential Operators and their Indices

163

Proof. Since the range R(Am ) is closed in L2 (M ), it follows from an application of the closed range theorem (see [100, Theorem 3.16], [147, p. 205, Theorem]) that (4.36) L2 (M ) = N (A∗m ) ⊕ R (Am ) . However we have, by formulas (4.33) and (4.30), N (A∗m ) = N (A∗ ), R(Am ) ∩ C ∞ (M ) = R(A).

(4.37a) (4.37b)

Therefore, the desired orthogonal decomposition (4.35) follows from assertions (4.37), by restricting the orthogonal decomposition (4.36) to the space C ∞ (M ). The proof of Lemma 4.57 is complete.   Now we can prove the following theorem: Theorem 4.58. If A ∈ Lm cl (M ) is elliptic, then we have, for all s ∈ R, ind As = dim N (A) − codim R(A). Here

(4.38)

codim R(A) = dim C ∞ (M )/R(A).

Proof. The orthogonal decomposition (4.35) tells us that dim N (A∗ ) = codim R(A). Hence, the desired formula (4.38) follows from formula (4.32). The proof of Theorem 4.58 is complete.   We let ind A = dim N (A) − dim N (A∗ ) = dim N (A) − codim R(A).

(4.39)

The next theorem states that the index of an elliptic pseudo-differential operator depends only on its principal symbol: Theorem 4.59. If A, B ∈ Lm cl (M ) are elliptic and if they have the same homogeneous principal symbol, then it follows that ind A = ind B.

(4.40)

Proof. Since the difference A − B belongs to the class Lm−1 (M ), it follows cl from Rellich’s theorem that the operator As − Bs : H s (M ) −→ H s−m (M )

164

4 Lp Theory of Pseudo-Differential Operators

is compact. Hence, by applying [100, Theorem 7.8] (cf. [80, Theorem 2.26]) we obtain that ind As = ind (Bs + (As − Bs )) = ind Bs . In view of Theorem 4.58, this proves the desired formula (4.40). The proof of Theorem 4.59 is complete.   As for the product of elliptic pseudo-differential operators, we have the following theorem: 

m Theorem 4.60. If A ∈ Lm cl (M ) and B ∈ Lcl (M ) are elliptic, then we have the formula ind BA = ind B + ind A. (4.41)

Proof. We remark that we have, for each s ∈ R, (BA)s = Bs−m · As . Hence, by applying [100, Theorem 7.3] (cf. [80, Theorem 2.21]) to our situation we obtain that ind (BA)s = ind Bs−m + ind As . This proves the desired formula (4.41), since BA is an elliptic operator in  Lm+m (M ) (cf. Theorem 4.42). cl The proof of Theorem 4.60 is complete.   As for the adjoints, we have the following theorem: Theorem 4.61. If A ∈ Lm cl (M ) is elliptic, then we have the formula ind A∗ = −ind A.

(4.42)

Indeed, it suffices to note that A∗∗ = A. We give some useful criteria for ind A = 0: ∗ Theorem 4.62. If A ∈ Lm cl (M ) is elliptic and if A and A have the same homogeneous principal symbol, then it follows that

ind A = 0.

(4.43)

Proof. Theorem 4.59 tells us that ind A = ind A∗ . However, in view of formula (4.42) this implies the desired formula (4.43). The proof of Theorem 4.62 is complete.   Corollary 4.63. If A ∈ Lm cl (M ) is elliptic and if its homogeneous principal symbol is real, then we have the assertion ind A = 0.

4.6 Elliptic Pseudo-differential Operators and their Indices

165

Indeed, by Theorem 4.41 it follows that A and A∗ have the same homogeneous real principal symbol. Therefore, Theorem 4.62 applies. ∗ Theorem 4.64. If A ∈ Lm cl (M ) is elliptic and if A = λA for some λ ∈ C, then we have the assertions |λ| = 1,

and ind A = 0. Proof. First, we remark that A = A∗∗ = (λA)∗ = λA∗ = |λ|2 A. Hence we have |λ| = 1 and so 

N (λA) = N (A), R(λA) = R(A).

Therefore, it follows from formula (4.42) that ind A = ind λA = ind A∗ = −ind A. This proves that ind A = 0. The proof of Theorem 4.64 is complete.   The next theorem describes conditions under which an elliptic pseudodifferential operator is invertible on Sobolev spaces: Theorem 4.65. Let A ∈ Lm cl (M ) be elliptic. Assume that  ind A = 0, N (A) = {0}. Then we have the following three assertions (i), (ii) and (iii): (i) The operator A : C ∞ (M ) → C ∞ (M ) is bijective. (ii) The operator As : H s (M ) → H s−m (M ) is an isomorphism for each s ∈ R. (iii) The inverse A−1 of A belongs to the class L−m cl (M ). Proof. (i) Since ind A = 0 and N (A) = {0}, it follows from formula (4.39) that N (A∗ ) = {0}. Hence the surjectivity of A follows from the orthogonal decomposition (4.35). (ii) Since N (As ) = N (A) = {0} and ind As = ind A = 0, it follows that the operator As : H s (M ) −→ H s−m (M )

166

4 Lp Theory of Pseudo-Differential Operators

is bijective for each s ∈ R. Therefore, by applying the closed graph theorem (see [100, Theorem 4.10], [147, p. 79, Theorem 1]) to our situation we obtain that the inverse As −1 : H s−m (M ) −→ H s (M ) is continuous for each s ∈ R. (iii) Since we have the formula  A−1 = As −1 C ∞ (M) and since each As −1 : H s−m (M ) → H s (M ) is continuous, it follows that the operator A−1 : C ∞ (M ) −→ C ∞ (M ) is continuous, and also it is of order −m. The situation can be visualized as in Figure 4.26 above. A −1

H s−m (M ) − −−s−− → H s (M )

C ∞ (M )

− −−−− → C ∞ (M ) A−1

Fig. 4.26. The mapping properties of the inverses A−1 and As −1 −m It remains to prove that A−1 ∈ L−m cl (M ) takes a parametrix B ∈ Lcl (M ) for A:  AB = I + P, P ∈ L−∞ (M ), BA = I + Q, Q ∈ L−∞ (M ).

Then we have the formula A−1 − B = (I − BA) A−1 = −Q · A−1 . However, in view of assertion (4.25) it follows that the operator Q · A−1 ∈ L−∞ (M ), since it is of order −∞. This proves that A−1 = B − Q · A−1 ∈ L−m cl (M ). The proof of Theorem 4.65 is complete.   The next theorem states that the Sobolev spaces H s (M ) can be characterized in terms of elliptic pseudo-differential operators:

4.6 Elliptic Pseudo-differential Operators and their Indices

167

Theorem 4.66. Let A ∈ Lm cl (M ) be elliptic with m > 0. Assume that  A = A∗ , N (A) = {0}. Then we have the following two assertions (i) and (ii): (i) There exists a complete orthonormal system {ϕj } of L2 (M ) consisting of eigenfunctions of A, and its corresponding eigenvalues {λj } are real and |λj | → +∞ as j → ∞. (ii) A distribution u ∈ D (M ) belongs to H mr (M ) for some integer r if and only if we have the condition ∞ 

2

λ2r j |(u, ϕj )| < +∞.

j=1

More precisely, the quantity (u, v)mr =

∞ 

λ2r j (u, ϕj ) (v, ϕj )

(4.44)

j=1

is an admissible inner product for the Hilbert space H mr (M ). Proof. (i) Since A = A∗ , it follows from Theorem 4.61 that ind A = 0. Hence, by applying Theorem 4.65 we obtain that the operator A : C ∞ (M ) −→ C ∞ (M ) is bijective, and its inverse A−1 is an elliptic operator in L−m cl (M ). We let 7 −1 = the composition of (A−1 ) : L2 (M ) → H m (M ) A 0 m and the injection: H (M ) → L2 (M ). Then it follows from Rellich’s theorem that the operator 7 −1 : L2 (M ) −→ L2 (M ) A is compact. Furthermore, since C ∞ (M ) is dense in L2 (M ) and A = A∗ , we have the formula     7 7 −1 u, v = u, A −1 v for u, v ∈ L2 (M ), A 7 −1 where (·, ·) is the inner product of L2 (M ). This implies that the operator A is self-adjoint. Also, we have the assertion     −1 7 −1 = N (A−1 ) ) = {0}, N A 0 = N (A

4 Lp Theory of Pseudo-Differential Operators

168

since A−1 ∈ L−m cl (M ) is elliptic. Therefore, by applying the Hilbert–Schmidt 7 −1 we obtain theorem (see [123, Theorem 2.56]) to the self-adjoint operator A 2 that there exists a complete orthonormal system {ϕj } of L (M ) consisting 7 −1 , and its corresponding eigenvalues {μ } are real and of eigenfunctions of A j converges to zero as j → ∞. Since the eigenvalues μj are all non-zero, it follows that ϕj = However, note that and that

1 7 1  −1  A ϕ ∈ H m (M ). A−1 ϕj = 0 j μj μj  −1      A = A−1 m 0 H m (M)  −1  A : H m (M ) −→ H 2m (M ). m

Hence, we have the assertion ϕj =

1  −1  A ϕ ∈ H 2m (M ). m j μj

Continuing this way, we obtain that , ϕj ∈ H km (M ) = C ∞ (M ). k∈N

Therefore, we have the formula A−1 ϕj = μj ϕj , and so Aϕj = λj ϕj , with |λj | =

λj =

1 , μj

1 −→ +∞ as j → ∞. |μj |

(ii) For each integer r, we let  Ar =

Ar  −1 |r| A

if r ≥ 0, if r < 0,

where A0 = I. Then it follows that Ar is an elliptic operator in Lmr cl (M ) and that N (Ar ) = {0}. Moreover we have, by Theorem 4.60,

4.6 Elliptic Pseudo-differential Operators and their Indices

 r

ind A =

r ind A = 0   |r| ind A−1 = 0

169

if r ≥ 0, if r < 0.

Therefore, by applying Theorem 4.65 we obtain that the operator (Ar )mr : H mr (M ) −→ L2 (M ) is an isomorphism. Thus the quantity (u, v)mr = ((Ar )mr u, (Ar )mr v) is an admissible inner product for the Hilbert space H mr (M ). Furthermore, since {ϕj } is a complete orthonormal system of L2 (M ), we have, by Parseval’s formula, (u, v)mr =

∞ 

((Ar )mr u, ϕj ) ((Ar )mr v, ϕj ).

(4.45)

j=1

However, we have, by formula (4.29),  ∗ ∗ ((Ar )mr ) = (Ar ) 0 = (Ar )0 , since A = A∗ . Hence, it follows that ((Ar )mr u, ϕj ) = (u, (Ar )0 ϕj ) = (u, Ar ϕj ) = λrj (u, ϕj ) .

(4.46)

Therefore, the desired formula (4.44) follows from formulas (4.45) and (4.46). The proof of Theorem 4.66 is complete.   As one of the important applications of Theorem 4.66, we can obtain the following theorem: Theorem 4.67. Let Δ be the Laplace–Beltrami operator on M and let {χj } be the orthonormal system of L2 (M ) consisting of eigenfunctions of −Δ and {λj } its corresponding eigenvalues: −Δχj = λj χj ,

λj ≥ 0.

Then the functions χj span the Sobolev spaces H s (M ) for each s ∈ R. More precisely, the quantity (u, v)s =

∞ 

s

(1 + λj ) (u, χj ) (v, χj )

j=1

is an admissible inner product for the Hilbert space H s (M ).

170

4 Lp Theory of Pseudo-Differential Operators

4.7 Functional Calculus for the Laplacian via the Heat Kernel The heat kernel Kt (x) =

|x|2 1 e− 4t n/2 (4πt)

for t > 0

(4.47)

has many important and interesting applications in partial differential equations. Physically, the heat kernel Kt (x) expresses a thermal distribution of position x at time t in a homogeneous isotropic medium Rn with unit coefficient of thermal diffusivity, given that the initial thermal distribution is the Dirac measure δ(x) (see Figure 4.27 above).

t>0

t=0

Kt (x) Dirac measure • δ(x)

After t x

0

0

x

Fig. 4.27. An intuitive meaning of the heat kernel Kt (x)

In this section we derive Newtonian, Riesz and Bessel potentials in Example 4.7 and also the Poisson kernel for the Dirichlet boundary value problem, by calculating various convolution kernels for the Laplacian. (I) First, it is easy to verify that the function  Kt (x − y)f (y) dy for f ∈ S(Rn ), Kt ∗ f (x) = Rn

is a solution of the initial value problem for the heat equation  ⎧ ⎨ ∂ − Δ u(x, t) = 0 in Rn × (0, ∞), ∂t ⎩ u(x, 0) = f (x) on Rn . Therefore, we have the formal representation formula K t ∗ f = et Δ f

for t > 0.

(II) Next, we consider the Laplace transform:  ∞ e−st φ(t) dt, Lφ(s) = 0

(4.48)

4.7 Functional Calculus for the Laplacian via the Heat Kernel

171

where φ(s) is a function to be chosen later on. If we make the formal change of variables s := −Δ, then we obtain that



∞

(et Δ f )φ(t) dt   ∞  Kt (x − y)f (y) dy φ(t) dt = 0 Rn    ∞ = Kt (x − y) φ(t) dt f (y) dy.

Lφ(−Δ)f =

0

Rn

0

This implies that the convolution kernel of the operator Lφ(−Δ) is equal to the following:



∞

Kt (x) φ(t) dt.

0

(A) First, we let φ(t) := tβ−1 Since we have the formula   ∞ e−st φ(t) dt = Lφ(s) = 0

for Re β > 0. ∞

e−st tβ−1 dt = Γ (β) s−β ,

(4.49)

0

it follows from formula (4.49) that the convolution kernel of the operator (−Δ)

−β

is equal to the following: 1 Γ (β)

 0

∞

Kt (x) tβ−1 dt.

(4.50)

However, we have the following claim: Claim 4.68. Let 0 < Re β < n/2. We have, for x = 0,  ∞ 1 1 Γ (n/2 − β) Kt (x) tβ−1 dt = . Γ (β) 0 Γ (β) 4β π n/2 |x|n−2β Proof. By formula (4.47), we obtain that  ∞ 2 1 1 − |x| 4t tβ−1 dt e Γ (β) 0 (4πt)n/2

(4.51)

4 Lp Theory of Pseudo-Differential Operators

172

 ∞ |x|2 1 dτ 1 τ n/2 τ 1−β e− 4 τ 2 n/2 Γ (β) (4π) τ 0 ∞ |x|2 1 1 = τ n/2−β−1 e− 4 τ dτ Γ (β) (4π)n/2 0  n/2−β−1  ∞ 4σ 1 1 4 −σ = e dσ Γ (β) (4π)n/2 0 |x|2 |x|2  ∞ 1 1 1 1 e−σ σ n/2−β−1 dσ = Γ (β) π n/2 4β |x|n−2β 0 1 Γ (n/2 − β) = . Γ (β) 4β π n/2 |x|n−2β =

The proof of Claim 4.68 is complete.   By combining formulas (4.50) and (4.51), we have proved the following theorem (see Table 4.3 with α := 2 Re β): Theorem 4.69. Let 0 < Re β < n/2. The convolution kernel of the operator (−Δ)−β is equal to the following: Γ (n/2 − β) 1 n−2β β n/2 |x| Γ (β) 4 π

for x = 0.

Example 4.9 (the Newtonian potential). If n ≥ 3 and β = 1, then it follows from an application of Theorem 4.53 that the convolution kernel of the operator −1 (−Δ) is equal to the following: Γ (n/2 − 1) 1 . 4π n/2 |x|n−2 However, we remark that Γ (n/2 − 1) = ωn = so that

2π n/2 Γ (n/2)

n 2 Γ , n−2 2 (the surface area of the unit ball),

Γ (n/2 − 1) 1 = . (n − 2)ωn 4π n/2

Hence it follows that the convolution kernel of the operator

4.7 Functional Calculus for the Laplacian via the Heat Kernel

173

−1

(−Δ) is equal to the following:

1 1 . (n − 2)ωn |x|n−2 This is the Newtonian potential (see Table 4.3). Example 4.10 (the Riesz potential). If β = α/2 and 0 < Re α < n, then it follows from an application of Theorem 4.53 that the convolution kernel of the operator (−Δ)−α/2 is equal to the following: Γ ((n − α)/2) 1 . Γ (α/2) 2α π n/2 |x|n−α This is the Riesz potential of order α. (B) Secondly, if we replace s by s + 1 in formula (4.49) we obtain that  ∞ 1 −β e−(s+1)t tβ−1 dt (4.52) (s + 1) = Γ (β) 0  ∞ 1 = e−st e−t tβ−1 dt Γ (β) 0   1 L e−s sβ−1 . = Γ (β) Therefore, by taking s := −Δ, β := α/2 for Re α > 0, we have, by formula (4.24), Theorem 4.70 (the Bessel potential). If Re α > 0, then the convolution kernel of the operator (I − Δ)−α/2 is equal to the following: 1 1 Γ (α/2) (4π)n/2



∞

e−t−

|x|2 4t

t

α−n 2

0

dt . t

This is the Bessel potential of order α (see Table 4.3). If we let 1 1 Gα (x) = Γ (α/2) (4π)n/2



then we have the following claim:

0

∞

e−t−

|x|2 4t

t

α−n 2

dt t

for Re α > 0,

4 Lp Theory of Pseudo-Differential Operators

174

Claim 4.71. For Re α > 0, we have the assertion Gα ∈ L1 (Rn ). Proof. We remark that 1 (4πt)n/2 so that

 Rn



e−

|x|2 4t

dx = 1,

Rn

 ∞ 1 e−t t(Re α)/2−1 dt |Γ (α/2)| 0 Γ (Re α/2) . = |Γ (α/2)|

|Gα (x)| dx =

The proof of Claim 4.71 is complete.   Therefore, by Young’s inequality it follows that    −α  (I − Δ) f  = Gα ∗ f p p

≤

Γ (Re α/2)

f p |Γ (α/2)|

for f ∈ Lp (Rn ).

(C) Thirdly, if we let β2 1 φ(s) := √ e− 4s πs

for β > 0,

then we have the formula (see formula (6.8) in Claim 6.1)  ∞ β2 1 −β e = e−s √ e− 4s ds. πs 0 Indeed, by the residue theorem it follows that  iβτ   ∞ iβ τ e e dτ = 2πi Res = π e−β . 2 2 1 + τ 1 + τ −∞ τ =i Hence we obtain the desired formula (4.53)  1 ∞ ei β τ −β e = dτ π −∞ 1 + τ 2   ∞  ∞ 1 iβ τ −(1+τ 2 )s = e e ds dτ π −∞ 0   ∞  ∞ 1 −s i β τ −s τ 2 = e e e dτ ds π 0 −∞

(4.53)

4.7 Functional Calculus for the Laplacian via the Heat Kernel

=

1 π 

= 0



∞

0 ∞

e−s

8

π − β2 e 4s s

175

 ds

β2 1 e−s √ e− 4s ds. πs

Now, by taking β := t

√ −Δ,

in formula (4.53), we have the formal representation formula  ∞ √ 1  t2 Δ  e−t −Δ f = e−s √ e 4s f ds πs 0   ∞ −s  e √ = Kt2 /4s (x − y) f (y) dy ds πs 0 Rn    ∞ −s e √ Kt2 /4s (x − y) ds f (y) dy. = πs 0 Rn This implies that the convolution kernel of the operator e−t

√ −Δ

is equal to the following:  0

∞

e−s √ Kt2 /4s (x) ds. πs

(4.54)

However, we have the following claim: Claim 4.72. We have, for t > 0,  ∞ −s e t Γ ((n + 1)/2) √ Kt2 /4s (x) ds = . (n+1)/2 2 + t2 )(n+1)/2 πs π (|x| 0

(4.55)

Proof. Indeed, by using formula (4.47) we have the desired formula (4.55)  ∞ −s e √ Kt2 /4s (x), ds πs 0  ∞ −s 2s 1 e − |x| t2 √ e ds = πs (πt2 /s)n/2 0  ∞ |x|2 1 1 e−s(1+ t2 ) s(n−1)/2 ds = (n+1)/2 n t 0 π  −(n+1)/2  ∞ 1 |x|2 1 = (n+1)/2 n 1 + 2 e−σ σ (n+1)/2−1 dσ t t π 0 t Γ ((n + 1)/2) . = π (n+1)/2 (|x|2 + t2 )(n+1)/2 The proof of Claim 4.72 is complete.  

4 Lp Theory of Pseudo-Differential Operators

176

By combining formulas (4.54) and (4.55), we have proved the following theorem: Theorem 4.73 (the Poisson kernel). The convolution kernel of the operator √ e−t −Δx is equal to the following: t Γ ((n + 1)/2) . (n+1)/2 2 π (|x| + t2 )(n+1)/2 This is called the Poisson kernel. More precisely, we can prove that the function u(x, t) = e−t

√

−Δx

f (x) 

Γ ((n + 1)/2) = π (n+1)/2

Rn

(|x −

y|2

t f (y) dy + t2 )(n+1)/2

is a solution of the following Dirichlet problem  ⎧ 2 ⎨ ∂ + Δ u(x, t) = 0 in Rn+1 , x + ∂t2 ⎩ u(x, 0) = f (x) on ∂Rn+1 = Rn , + where

  Rn+1 = (x, t) ∈ Rn+1 : t > 0 . +

4.8 Notes and Comments Sections 4.1 through 4.3: The function spaces discussed here are adapted from Adams–Fournier [2], Bergh–L¨ofstr¨om [13], Calder´ on [21], Friedman [43], Taibleson [112] and Triebel [135]. Schwartz [102] and Gelfand–Shilov [49] are the classics for distribution theory. Our treatment in this book follows the expositions of Chazarain–Piriou [26], H¨ormander [56] and Treves [134]. Section 4.3: This section is a modern version of trace and sectional trace theorems in terms of pseudo-differential operators. Sections 4.4 and 4.5: For detailed studies of Fourier integral operators and pseudo-differential operators, the reader is referred to Chazarain–Piriou [26], Duistermaat [31], Duistermaat–H¨ormander [32], H¨ormander [60], [61], Kumano-go [73], Rempel–Schulze [93] and Taylor [133]. Section 4.6: This section is devoted to a modern Fredholm operator theory of elliptic pseudo-differential operators on a manifold, which is adapted from Palais [86, Chapter VII] and Taira [114, Section 8.7]. Gohberg–Kre˘ın [51] is

4.8 Notes and Comments

177

the classic for index theory of linear Fredholm operators in Banach spaces. See also McLean [80, Chapter 2] and Schechter [100, Chapter 7]. Section 4.7: The material is adapted from Folland [41, Chapter 4, Section B]. This chapter is an expanded and revised version of Chapter 3 of the second edition [121].

5 Boutet de Monvel Calculus

In this chapter we introduce the notion of transmission property due to Boutet de Monvel [17], [18] and [19], which is a condition about symbols in the normal direction at the boundary. Elliptic boundary value problems cannot be treated directly by pseudo-differential operator methods. It was Boutet de Monvel [19] who brought in the operator-algebraic aspect with his calculus in 1971. He constructed a relatively small “algebra”, called the Boutet de Monvel algebra, which contains the boundary value problems for elliptic differential operators as well as their parametrices. We take a close look at Boutet de Monvel’s work. Let Ω be a bounded, domain of Euclidean space Rn with smooth boundary ∂Ω. Without loss of generality, we may assume that Ω is a relatively compact,  without open subset of an n-dimensional, compact smooth manifold M = Ω boundary (see Figure 5.1 below). The manifold M is called the double of Ω (see [83]).

Ω

M=Ω

∂Ω

 of Ω Fig. 5.1. The double M = Ω

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 5

180

5 Boutet de Monvel Calculus

If we study the Dirichlet problem in the domain Ω, it is natural to use the zero extension e+ u of functions u outside of the closure Ω = Ω ∪ ∂Ω:  u(x) for x ∈ Ω, + e u(x) = 0 for x ∈ M \ Ω. The zero extension has a probabilistic interpretation. Namely, this corresponds to stopping the diffusion process with jumps into the double M at the first exit time of the closure Ω. Boutet de Monvel [19] introduced a 2 × 2 matrix of operators ⎞ ⎛ PΩ + G K C ∞ (Ω) C ∞ (Ω) ⎠: A=⎝ −→ T S C ∞ (∂Ω) C ∞ (∂Ω) Here: (1) P is a pseudo-differential operator on the full manifold M and    PΩ u = r+ P (e+ u) = P (e+ u)Ω for all u ∈ C ∞ (Ω), where e+ u is the extension of u by zero to M and r+ v is the restriction v|Ω to Ω of a distribution v on M . In view of the pseudo-local property of P , we find that the operator PΩ can be visualized as follows: e+

r+

PΩ : C ∞ (Ω) −→ D (M ) −→ D (M ) −→ C ∞ (Ω). P

The crucial requirement is that the symbol of P has the transmission property in order that PΩ maps C ∞ (Ω) into itself. In Section 5.1 we introduce three basic function spaces H, H + and H − (Proposition 5.1). In Section 5.2 we illustrate how the transmission property of the symbol ensures that the associated operator preserves smoothness up to the boundary (Theorems 5.6 and 5.7). (2) S is a pseudo-differential operator on ∂Ω. (3) The potential operator K and trace operator T are generalizations of the potentials and trace operators known from the classical theory of elliptic boundary value problems, respectively. (4) The entry G, a singular Green operator, is an operator which is smoothing in the interior Ω while it acts like a pseudo-differential operator in directions tangential to the boundary ∂Ω. As an example, we may take the difference of two solution operators to (invertible) classical boundary value problems with the same differential part in the interior but different boundary conditions. In Section 5.3 we give typical examples of a potential operator K, a trace operator T and a singular Green operator G (see Examples 5.5 through 5.12).

5.1 The Spaces H, H + and H −

181

Boutet de Monvel [19] proved that these operator matrices form an algebra in the following sense (see [93, p. 175, Theorem 1 and p. 195, Theorem 2]): Given another element of the calculus, say, ⎞ ⎛  PΩ + G K  C ∞ (Ω) C ∞ (Ω)  ⎠ ⎝ : A = −→ T S C ∞ (∂Ω) C ∞ (∂Ω) the composition A A is again an operator matrix of the type described above. It is worth pointing out here that the product PΩ PΩ does not coincide with (P  P )Ω ; in fact, the difference PΩ PΩ − (P  P )Ω turns out to be a singular Green operator (see [93, p. 100, Lemma 5]). A general symbol class of the Boutet de Monvel calculus except for the directions tangential to ∂Ω may be summarized as in Table 5.1 below.

Operators

Pseudo-differential operators with transmission property

Singular Green operators

Potential operators

Trace operators

Notation

PΩ

G

K

T

Symbol classes

H = H+ ⊕ H−

H+ ⊗ H−

H+

H − = H0− ⊕ H

Table 5.1. A general symbol class of the Boutet de Monvel calculus

It should be emphasized that the Boutet de Monvel calculus is closely related to the classical Wiener–Hopf technique (see [145], [84]) that remains an extremely important tool for modern scientists, and the areas of application continue to broaden. The last Section 5.4 is devoted to a brief historical perspective of the Wiener–Hopf technique.

5.1 The Spaces H, H + and H − In this section we introduce three basic function spaces H, H + and H − (Proposition 5.1). We let S (R) = the Schwartz space of all rapidly decreasing functions on R. The dual space is the space S  (R) of tempered distributions on R. The Fourier transform

182

5 Boutet de Monvel Calculus

F : S (R) −→ S (R) will in general be indicated by a hat: For u ∈ S (R), we let  e−i x·ξ u(x) dx for ξ ∈ R. u (ξ) = (F u)(ξ) = R

Given a distribution u on R, we write r+ u for its restriction to the right half-axis R+ : r+ u = u|R+ , where R+ = {x ∈ R : x > 0} . −

Similarly, we write r u for its restriction to the left half-axis R− : r− u = u|R− , where R− = {x ∈ R : x < 0} . We let     S R+ = r+ u : u ∈ S(R) ,     S R− = r− u : u ∈ S(R) , and Sθ (R) = S(R+ ) ⊕ S(R+ ). Given a function f on R+ , we denote by e+ f its extension by zero to a function on R:   +  f (x) if x ∈ R+ , e f (x) = 0 if x ∈ R \ R+ . Similarly, given a function g on R− , we denote by e− g its extension by zero to a function on R:   −  g(x) if x ∈ R− , e g (x) = 0 if x ∈ Rn \ R− , where R− = {x ∈ R : x < 0}. We let     H + := F S(R+ ) = (e+ u)∧ : u ∈ S(R+ ) ,     H0− := F S(R− ) = (e− u)∧ : u ∈ S(R− ) , and

H0 = H + ⊕ H0− .

5.1 The Spaces H, H + and H −

183

We remark that H + and H0− are spaces of smooth functions on R decaying to the first order near infinity. We consider H + and H0− in the image topology of S(R+ ) and S(R− ), respectively. Then the Fourier transform F : Sθ (R) = S(R+ ) ⊕ S(R+ ) −→ H0 = H + ⊕ H0− is an isomorphism of Fr´echet spaces. Now we define the continuous projections Π ± := F r± F −1 as follows: F −1

r+

F

F −1

r−

F

Π + : H0 −→ Sθ (R) −→ S(R+ ) −→ H + , Π − : H0 −→ Sθ (R) −→ S(R− ) −→ H0− . We denote by H  the space of all polynomials, and let H − = H0− ⊕ H  , H = H + ⊕ H0− ⊕ H  = H + ⊕ H − . We extend the projections Π ± to the space H = H0 ⊕ H  by setting  Π + = 0 on H  , Π − = I on H  , so that

Π− = I − Π+

on H.

We give simple but important examples of elements of H + and H − : Example 5.1. Let λ be a positive number. For any non-negative integer k, the function (λ − it)k (λ + it)k+1 is the Fourier transform of the product of a Laguerre polynomial by an exponential: ⎧  k  k ⎪ d x −λ x ⎨ λ− e if x > 0, ϕk (x) = dx k! ⎪ ⎩ 0 if x < 0. In a similar way, a function f (ξ) ∈ H − is the Fourier transform of the sum of a function ϕk (−x), and of a linear combination of derivatives of the Dirac measure at the origin. Part (a) of the following Proposition 5.1 is easily verified. Parts (b), (c) and (d) provide Paley–Wiener type characterizations of the spaces H + , H0− and H0 , respectively (see Rempel–Schulze [93, Chapter 2, Subsection 2.1.1]):

184

5 Boutet de Monvel Calculus

Proposition 5.1. (a) The spaces H + , H0− , H − , H0 and H are algebras. (b) A function h(t) ∈ C ∞ (R) belongs to H + if and only if it has an analytic extension h(ζ) to the lower open half-plane {Im ζ < 0}, continuous in the lower closed half-plane {Im ζ ≤ 0}, together with an asymptotic expansion h(ζ) ∼

∞ 

ak ζ −k ,

(5.1)

k=1

for |ζ| → ∞ in the lower closed half-plane {Im ζ ≤ 0}, which can be differentiated formally (a complex analytic version). A function h(t) ∈ C ∞ (R) belongs to H + if and only if it has a unique expansion h(t) =

∞ 

αk

k=0

(1 − it)k 1 1 − it + α1 = α0 + ..., (1 + it)k+1 1 + it (1 + it)2

where the coefficients αk form a rapidly decreasing sequence (a real analytic version). (c) A function h(t) ∈ C ∞ (R) belongs to H0− if and only if it has an analytic extension h(ζ) to the upper open half-plane {Im ζ > 0}, continuous in the upper closed half-plane {Im ζ ≥ 0}, together with an asymptotic expansion h(ζ) ∼

∞ 

ak ζ −k ,

(5.2)

k=1

for |ζ| → ∞ in the upper closed half-plane {Im ζ ≥ 0}, which can be differentiated formally (a complex analytic version). A function h(t) ∈ C ∞ (R) belongs to H0− if and only if it has a unique expansion h(t) =

−∞  k=−1

αk

(1 + it)|k|−1 1 1 + it + α−2 = α−1 + ..., 1 − it (1 − it)2 (1 − it)|k|

where the coefficients αk form a rapidly decreasing sequence (a real analytic version). (d) h(t) ∈ C ∞ (R) belongs to H0 = H + ⊕ H0− if and only if it has an expansion ∞  ak ζ −k , h(ζ) ∼ k=1

for |ζ| → ∞ in R, which can be differentiated formally (a complex analytic version). A function h(t) ∈ C ∞ (R) belongs to H0 if and only if it has a unique expansion

5.2 Transmission Property of Pseudo-Differential Operators

h(t) =

∞  k=−∞

αk

185

(1 − it)k , (1 + it)k+1

where the coefficients αk form a rapidly decreasing sequence (a real analytic version). Proposition 5.2. Assume that h(t) is analytic on the real line R and meromorphic at infinity. If h(ζ) is a meromorphic extension of h to the whole complex plane C, then we let   + h(t) dt = h(ζ) dζ, (5.3) γ

where γ is a large circle in the upper open half-plane {Im ζ > 0}. Then we have the following two assertions (i) and (ii): ++ f (t) dt = 0. (i) If f (t) ∈ H − = H0− ⊕ H  , it follows that (ii) If h(t) ∈ H + is integrable on R, we have the formula  ∞  + h(t) dt = h(t) dt = 0. −∞

Remark 5.3. Assume that f (t) and g(t) are analytic on the real line R and meromorphic at infinity. By Proposition 5.2, we have the formula + +  + f (t) · Π − g(t) dt if f ∈ H + , f (t)g(t) dt = + + f (t) · Π + g(t) dt if f ∈ H − .

5.2 Transmission Property of Pseudo-Differential Operators Given an arbitrary pseudo-differential operator A, it is in general not true that the operator    AΩ : u −→ r+ A(e+ u) = A(e+ u)Ω maps functions u which are smooth up to the boundary ∂Ω into functions AΩ u with the same property. The crucial requirement here is that the symbol of A has the transmission property. On one hand, this restricts the class of boundary value problems in the calculus, on the other hand, however, it ensures that solutions to elliptic equations with smooth data are smooth; it therefore helps to avoid problems with singularities of solutions at the boundary. In this section, following Boutet de Monvel [17], [18] and [19] we introduce the notion of transmission property which is a condition about symbols in the normal direction at the boundary in order to ensure the stated regularity property (see Rempel–Schulze [93]).

186

5 Boutet de Monvel Calculus

If x = (x1 , x2 , . . . , xn ) is the variable in Rn , we write x = (x , xn ) ,

x = (x1 , x2 , . . . , xn−1 ),

so x ∈ Rn−1 is the tangential component of x with dual variables ξ  = (ξ1 , ξ2 , . . . , ξn−1 ) , and xn ∈ R is its normal component with dual variable ξn . If m ∈ R, we let  n  m m R+ × Rn = the space of symbols a(x, ξ) in S1,0 (Rn+ × Rn ) S1,0 m which have an extension in S1,0 (Rn × Rn ).

Now we are in a position to define the transmission property of symbols m (Rn+ × Rn ): in the class S1,0 m (Rn+ × Rn ) is said to have the transDefinition 5.1. A symbol a(x, ξ) ∈ S1,0 mission property with respect to the boundary Rn−1 if all its derivatives γ  ∂ a(x , 0, ξ  , ν) for γ ≥ 0 ∂xn

admit an expansion of the form a(γ) (x , 0, ξ  , ν) (5.4)  γ ∂ a(x , 0, ξ  , ν) = ∂xn m ∞ −1     (1 − iν ξ   )k −1 j = bγ j (x , ξ  ) ν ξ   + aγ k (x , ξ  ) for ν ∈ R, −1 (1 + iν ξ   )k+1 j=0 k=−∞ where (cf. [93, p. 119, Proposition 3])   n−1 m bγ j (x , ξ  ) ∈ S1,0 Rn−1 , × R   x ξ and

  n−1 m Rn−1 × R aγ k (x , ξ  ) ∈ S1,0   x ξ

form a rapidly decreasing sequence with respect to k, and 1/2  ξ   = 1 + |ξ  |2 . It is easy to see that the expansion condition (5.4) is equivalent to the following: γ    ∂ n−1  m Rn−1 a(x , 0, ξ  , ξ   ν) ∈ S1,0 × R ⊗ π Hν .   x ξ ∂xn

5.2 Transmission Property of Pseudo-Differential Operators

187

m  π Hν is the completed π-topology (or Here the space S1,0 (Rn−1 × Rn−1 )⊗ m (Rn−1 × Rn−1 ) projective topology) tensor product of the Fr´echet spaces S1,0 and Hν (see Schaefer [99, Chapter III, Section 6], Treves [134, Chapter 45]). For classical pseudo-differential symbols of integer order, the transmission property can be expressed via homogeneity properties of the terms in the asymptotic expansion (see [17, Proposition (2.3.2)]): m (Rn × Rn ) is a classical symbol of order m ∈ Z Remark 5.4. If a(x, ξ) ∈ S1,0 with an asymptotic expansion

a(x, ξ) ∼

∞ 

am−j (x, ξ),

j=0 m−j where am−j (x, ξ) ∈ S1,0 (Rn ×Rn ) is positively homogeneous of degree m−j for |ξ| ≥ 1, then it is easy to verify (see [93, p. 123, Proposition 1], [61, Lemma 18.2.14]) that a(x, ξ) has the transmission property if and only if we have, for all multi-indices α = (α , αn ),



∂ ∂xn

αn 

∂ ∂ξ  

m−j−|α |

= (−1)

α

∂ ∂xn

am−j (x , 0, 0, +1) αn 

∂ ∂ξ 

α

(5.5)

am−j (x , 0, 0, −1).

It should be emphasized that this is no longer true for non-classical pseudodifferential symbols. For details, see the analysis by Grubb–H¨ormander [53]. We give two typical examples of classical symbols having the transmission property: Example 5.2. (1) The symbol a(x, ξ) of the second-order, elliptic differential operator n  ∂2 −Δ + 1 = − +1 ∂x2i i=1 is given by the formula (m = 2)   2 n−1  2 a(x, ξ) = |ξ|2 + 1 = ξ   + ν 2 ∈ S1,0 Rn−1 × R ⊗ π Hν   x ξ (2) The symbol a(x, ξ)−1 is given by the formula (m = −2) a(x, ξ)−1 = =

1 |ξ|2 + 1

  1 1 1 1 −2 n−1 n−1  + ∈ S × R R ⊗π Hν .   1,0 x ξ 2 ξ   ξ   + iν 2 ξ   ξ   − iν

We recall (Subsection 4.1.5) that the symbol

188

5 Boutet de Monvel Calculus

1 |ξ|2 + 1 corresponds to the Bessel potential of order 2 (see Aronszajn–Smith [12])  G2 ∗ u(x) = G2 (x − y)u(y) dy, Rn

and further that the function G2 (x) can be expressed in the form (see Stein [108, Chapter V, Section 3])  ∞ |x|2 2−n dt 1 G2 (x) = e−t− 4t t 2 n/2 t (4π) 0 1 1 = n n/2 K(n−2)/2 (|x|) (n−2)/2 , 2 π |x| where K(n−2)/2 (z) is the modified Bessel function of the third kind (see Watson [142]). We let m n n Lm 1,0 (R+ ) = the space of pseudo-differential operators in L1,0 (R+ )

which can be extended to a pseudo-differential operator n in the class Lm 1,0 (R ). n Now we can define pseudo-differential operators in the class Lm 1,0 (R ) to have the transmission property: n Definition 5.2. A pseudo-differential operator A ∈ Lm 1,0 (R+ ) is said to have the transmission property with respect to the boundary Rn−1 if any complete symbol of A has the transmission property with respect to the boundary Rn−1 .

As a first useful example, we formulate the pseudo-differential operators that have the transmission property in dimension one ([19, Theorem 2.7]): Theorem 5.5. Let A(x, D) be a pseudo-differential operator defined in a neighborhood of the half-line x ≥ 0. In order that the transmission property with respect to the origin holds true for A, it is necessary and sufficient that A admits a decomposition A = A0 + A1 + A2 . Here: (1) The symbol of A0 vanishes to the infinite order at the origin x = 0. (2) A1 is a differential operator with smooth coefficients. (3) The distribution kernel of A2 is a function f (x, y) which is smooth up to the diagonal for x > y, and also for x < y.

5.2 Transmission Property of Pseudo-Differential Operators

189

We give two simple examples A and B of pseudo-differential operators which have the transmission property (see Example 4.1): n Example 5.3. (i) The symbol a(x, ξ) of A ∈ Lm 1,0 (R ) is given by the formula 9 :   ξn2 n−1 n−1   m m a(x, ξ) = ξ  exp − R ∈ S × R ⊗π Hξn .   1,0 x ξ 2 ξ   n (ii) The symbol b(x, ξ) of B ∈ Lm 1,0 (R ) is given by the formula

b(x, ξ)  m

= ξ  Here

$ 1−

2ξn2

2 ξ  

ξ = (ξ  , ξn ),

%

9 exp −

:

ξn2

2 ξ  

  m  π Hξn . Rn−1 × Rn−1 ∈ S1,0 ⊗ x ξ

ξ  = (ξ1 , ξ2 , . . . , ξn−1 ),

ξ   =

; 1 + |ξ  |2 .

Now we illustrate how the transmission property of the symbol ensures that the associated operator preserves smoothness up to the boundary. If A n is a pseudo-differential operator in Lm 1,0 (R+ ), then we define a new operator  n   ∞ ARn+ : C(0) R+ −→ C ∞ Rn+  u −→ A(e+ u) n , R+

where

  ∞ C(0) (Rn+ ) = u|Rn : u ∈ C0∞ (Rn ) . +

The transmission property implies that if u is smooth up to the boundary, then so is ARn+ u. More precisely, we have the following theorem (see [26, Chapitre V, Th´eor`eme 2.5]; [93, p. 137, Corollary 3 and p. 168, Theorem 8]): n Theorem 5.6. Let A ∈ Lm 1,0 (R+ ). Then we have the following two assertions (i) and (ii) (see Figure 5.2 below): n−1 , (i) If A has the transmission property  with  respect to the boundary  nR  ∞ ∞ n n R+ . then the operator AR+ maps C(0) R+ continuously into C (ii) If A has the transmission property Rn−1 , then  nwith  respect to the boundary k−m+θ k+θ the operator ARn+ maps Ccomp R+ continuously into Cloc Rn+ for any integer k ≥ m and 0 < θ < 1. Here   n   k+θ n k+θ Ccomp R+ = C (R ) ∩ C0 (Rn ) Rn +   k+θ n = u|Rn : u ∈ C (R ) with compact support in Rn +

and

190

5 Boutet de Monvel Calculus k+θ Ccomp Rn +

∞ C(0) Rn +

ARn

+

k−m+θ −−−−− → Cloc Rn +

−−−−− → ARn

C ∞ Rn +

+

Fig. 5.2. The mapping properties of the operator ARn+

  n k−m+θ k−m+θ Cloc R+ = Cloc (Rn )Rn +   k−m+θ n = u|Rn : ϕ u ∈ C (R ) for all ϕ ∈ C0∞ (Rn ) . +

Moreover, it should be noticed that the notion of transmission property is invariant under a change of coordinates which preserves the boundary. Hence the notion of transmission property can be transferred to a bounded domain Ω of Euclidean space Rn with smooth boundary ∂Ω. Indeed, if Ω is a relatively  (see Figure 5.1), then the notion compact open subset of the double M = Ω of transmission property can be extended to the class Lm 1,0 (M ), upon using local coordinate systems flattening out the boundary ∂Ω. Then we have the following theorem (see [93, p. 139, Theorem 4 and p. 176, Theorem 1]): Theorem 5.7. Let A ∈ Lm 1,0 (M ). Then we have the following two assertions (i) and (ii) (see Figure 5.3 below): (i) If A has the transmission property with respect to the boundary ∂Ω, then the operator AΩ : C ∞ (Ω) −→ C ∞ (Ω)  u −→ A(e+ u)

Ω

maps C ∞ (Ω) continuously into itself, where e+ u is the zero extension of u to M by zero outside of Ω. (ii) If A has the transmission property with respect to the boundary ∂Ω, then the operator AΩ maps C k+θ (Ω) continuously into C k−m+θ (Ω) for any integer k ≥ m and 0 < θ < 1. Following Schrohe [101, Definition 2.3], we can introduce an equivalent definition of the transmission property for general symbols a(x, y, ξ) in the m (Rn × Rn × Rn ). To do this, let H be the linear space of all complexclass S1,0 valued functions f (t) on the real line R which are smooth and have a regular pole at infinity. More precisely, a function f (t) ∈ C ∞ (R) belongs to H if and only if it has a unique expansion

5.3 Trace, Potential and Singular Green Operators on Rn + C k+θ Ω

C∞ Ω

191

A

Ω − −−− −→ C k−m+θ Ω

− −−−−→ AΩ

C∞ Ω

Fig. 5.3. The mapping properties of the operator AΩ

f (t) =

N 

αs ts +

s=1

∞  k=−∞

 αk

1 − it 1 + it

k ,

i=

√ −1,

where the coefficients αk form a rapidly decreasing sequence (see [93, Chapter 2, Section 2.1.1]). Now we are in a position to define the transmission property of symbols m (Rn × Rn × Rn ): in the class S1,0 Definition 5.3. A symbol m a(x, y, ξ) = a(x , xn , y  , yn , ξ  , ξn ) ∈ S1,0 (Rnx ,xn × Rny ,yn × Rnξ ,ξn )

is said to have the transmission property at {xn = yn = 0}, provided that we have, for all non-negative integers k and ,   ∂ k+ a  m  π Hξn , Rn−1 (x , 0, y  , 0, ξ  , ξ   ξn ) ∈ S1,0 × Rn−1 × Rn−1 ⊗ x y ξ k  ∂xn ∂yn where

1/2  . ξ   = 1 + |ξ  |2

The subscripts x , y  , ξ  and ξn are used in order to indicate the variable for which we have the corresponding property.

5.3 Trace, Potential and Singular Green Operators on Rn + In this section we give basic definitions and properties of classes of trace, potential and singular Green operators on the half-space Rn+ . The presentation here is based on Rempel–Schulze [93, Chapter 2, Subsection 2.3.2]. Pseudo-differential trace operators T are a natural generalization of the usual differential trace operators from elliptic boundary value problems, while potential operators K can be described as the adjoints of trace operators T with respect to the L2 inner products. Singular Green operators G are introduced in order to get algebra of matrices of operators of the form

192

5 Boutet de Monvel Calculus

⎛ A=⎝

PΩ + G K T

⎞ ⎠,

S

called the Boutet de Monvel algebra. In fact, a typical example of a singular Green is the composition K ◦ T of a trace operator T and a potential operator K (see Example 5.11 below). A general symbol class of the operator A may be expressed schematically as follows: ⎛ ⎞ PΩ + G K ⎝ ⎠ T S ⎞ ⎛ ∗ − ∗  ∗   π Hν + S1,0  π Hd,τ S1,0 S1,0 ⊗ ⊗π Hν+ ⊗ ⊗π Hν+ ⎠. ⇐⇒ ⎝ − ∗  ∗ S1,0 S1,0 ⊗π Hd,τ Example 5.4. We give a concrete example of the symbol class of the operator A (see Examples 5.5 through 5.12): ⎛ ⎞ 1 1 1 1 + ξ   + iν ξ   − iτ ξ   + iν ⎟ ⎜ ξ  2 + ν 2 ⎜ ⎟ ⎜ ⎟. ⎝ ⎠ 1 1 ξ   − iτ ξ   Here and in the following we use the notation: ξ = (ξ  , ν) = (ξ1 , ξ2 , . . . , ξn−1 , ν) ∈ Rn , 6 2 ξ   = 1 + |ξ  | , ξ = (ξ  , ν) = (ξ1 , ξ2 , . . . , ξn−1 , ν) ∈ Rn ξ = (ξ  , τ ) = (ξ1 , ξ2 , . . . , ξn−1 , τ ) ∈ Rn

for potential operators K, for trace operators T .

5.3.1 Potential Operators on Rn + A function k (x , y  , ξ  , ν) ∈ C ∞ (Rn−1 × Rn−1 × Rn−1 × R) is called a potential symbol of order m if it satisfies the condition k(x , y  , ξ  , ν) =

∞  j=0

kj (x , y  , ξ  )

−1 j

(1 − iν ξ   (1 + iν

)

. ξ  −1 )j+1

(5.6)

Here the symbols kj (x , y  , ξ  ) form a rapidly decreasing sequence in the class m S1,0 (Rn−1 × Rn−1 × Rn−1 ). The condition (5.6) is equivalent to the following:

5.3 Trace, Potential and Singular Green Operators on Rn +

193

  m  π Hν+ . Rn−1 k(x , y  , ξ  , ξ   ν) ∈ S1,0 × Rn−1 × Rn−1 ⊗ x y ξ Then the potential operator K : C0∞ (Rn−1 ) −→ C ∞ (Rn+ ) is defined as an oscillatory integral by the formula

(Kv)(x , xn )     1 ˇ  , y  , ξ  , xn ) v(y  ) dy  dξ  = ei(x −y )·ξ k(x n−1 (2π) Rn−1 ×Rn−1 for all v ∈ C0∞ (Rn−1 ). Here:

ˇ  , y  , ξ  , xn ) = 1 k(x 2π



(5.7)

ei xn ν k(x , y  , ξ  , ν) dν.

R

If the potential symbol k(x , y  , ξ  , ν) does not depend on x and y  , then the oscillatory integral (5.7) reduces to the following:    1 ei(x ·ξ +xn ν) k(ξ  , ν) v(ξ  ) dξ  dν (Kv)(x , xn ) = n (2π) Rn for all v ∈ C0∞ (Rn−1 ). The mapping properties of the potential operator K of order m can be visualized as in Figure 5.4 below. s,p Bcomp Rn−1

C0∞ Rn−1

K

s−m−1+1/p,p

− −−−− → Hloc

− −−−− → K

Rn +

C ∞ Rn +

Fig. 5.4. The condition that 1 < p < ∞ and s ∈ R

Here

and

 n−1  s,p Bcomp R = B s,p (Rn−1 ) ∩ E  (Rn−1 )     = u ∈ B s,p Rn−1 : supp u is compact in Rn−1   s−m−1+1/p,p  n  s−m−1+1/p,p Hloc R+ = Hloc (Rn ) n R+   = u|Rn : ϕ u ∈ H s−m−1+1/p,p (Rn ) for all ϕ ∈ C0∞ (Rn ) . +

We give a typical example of potential operators:

194

5 Boutet de Monvel Calculus

Example 5.5 (Poisson operator). We let k (x , ξ  , ν) = =

1 iν + ξ  

  1 1 −1 n−1 n−1  R ∈ S × R ⊗π Hν+ ,   1,0 x ξ ξ   1 + iν ξ  −1

m = −1.

Then we have, by formula (5.3) and the residue theorem, ˇ  , ξ  , xn ) k(x   1 1 1 i xn ν dν = e = 2π R iν + ξ   2π  = e−xn ξ  ,

+

ei xn ν

1 dν i ν + ξ  

and the formula for the potential operator     1 ei x ·ξ e−xn ξ  v(ξ  ) dξ  (Kv)(x , xn ) = n−1 (2π) Rn−1 for all v ∈ C0∞ (Rn−1 ). We remark that the function u(x , xn ) = (Kv)(x , xn ) is a (unique) solution of the Dirichlet problem  Δu(x , xn ) = 0 in Rn+ , u(x , 0) = v(x ) on Rn−1 = ∂Rn+ , where Δ=

∂2 ∂2 ∂2 + 2 + ... + 2 2 ∂x1 ∂x2 ∂xn

is the usual Laplacian. Namely, the potential operator K is the Poisson operator or Poisson kernel for the Dirichlet problem (see formulas (5.1) and (5.2) in Section 5.1). 5.3.2 Trace Operators on Rn + A function t (x , y  , ξ  , τ ) ∈ C ∞ (Rn−1 × Rn−1 × Rn−1 × R) is called a trace symbol of order m and type d if it satisfies the condition t (x , y  , ξ  , τ ) =

d−1 

  −1 k bk (x , y  , ξ  ) τ ξ  

k=0 ∞ 

+

j=0

tj (x , y  , ξ  )

(5.8) −1 j

(1 + iτ ξ   (1 − iτ

)

. −1 ξ   )j+1

5.3 Trace, Potential and Singular Green Operators on Rn +

Here

195

m (Rn−1 × Rn−1 × Rn−1 ) bk (x , y  , ξ  ) ∈ S1,0

and the symbols tj (x , y  , ξ  ) form a rapidly decreasing sequence in the class m S1,0 (Rn−1 × Rn−1 × Rn−1 ). The condition (5.8) is equivalent to the following:   n−1 n−1  − m × R × R . Rn−1 ⊗π Hd,τ t (x , y  , ξ  , ξ   τ ) ∈ S1,0    x y ξ Then the trace operator ∞ T : C(0) (Rn+ ) −→ C ∞ (Rn−1 )

is defined as an oscillatory integral by the formula (T u)(x ) (5.9)     1 = ei(x −y )·ξ (2π)n−1 Rn−1 ×Rn−1      1 × t(x , y  , ξ  , τ ) e−i yn τ (e+ u)(y  , yn )dyn dτ dy  dξ  2π R R ∞ for all u ∈ C(0) (Rn+ ).

Here e+ u is the extension of u to Rn by zero outside of Rn+  u(y  , yn ) for yn ≥ 0, (e+ u)(y  , yn ) = 0 for yn < 0. If the trace symbol t(x , y  , ξ  , ν) does not depend on x and y  , then the oscillatory integral (5.9) reduces to the following (see Remark 5.3):     + 1 1 i x ·ξ    > + (T u)(x ) = e t(ξ , τ ) e u(ξ , τ ) dτ dξ  n−1 2π n−1 (2π) R ∞ for all u ∈ C(0) (Rn+ ).

We let

 n s,p Hcomp R+ := (H s,p (Rn ) ∩ E  (Rn ))|Rn +   s,p n n = u|R+ : u ∈ H (R ) with compact support in Rn

and s−m−1/p,p  n−1  Bloc R    n−1 ) : ψ u ∈ B s−m−1/p,p (Rn−1 ) for all ψ ∈ C0∞ (Rn−1 ) . := u ∈ D (R

Then the mapping properties of the trace operator T of order m and type d can be visualized as in Figure 5.5 below. We give three typical examples of trace operators:

196

5 Boutet de Monvel Calculus s,p Hcomp Rn +

∞ C(0) Rn +

T

s−m−1/p,p

−−−−− → Bloc

−−−−− → T

Rn−1

C ∞ Rn−1

Fig. 5.5. The condition that 1 < p < ∞, s − 1/p ∈ Z and s > d − 1 + 1/p

Example 5.6 (Dirichlet condition). We let   n−1  − 0 Rn−1 t0 (x , ξ  , τ ) = 1 ∈ S1,0 × R , ⊗π H1,τ   x ξ

m = 0, d = 1.

Then we have the formula    1 1 e−i yn τ (e+ u)(y  , yn ) dyn dτ 2π R R    1 i0τ = e e−i yn τ (e+ u)(y  , yn ) dyn dτ 2π R R ∞ (Rn+ ). = (e+ u)(y  , 0) = u(x , 0) for all u ∈ C(0)

Therefore, we have the formula for the trace operator (T0 u) (x )     1 = ei(x −y )·ξ n−1 (2π) Rn−1 ×Rn−1      1 × ei 0 τ e−i yn τ (e+ u)(y  , yn ) dyn dτ dy  dξ  2π R R     1 = ei(x −y )·ξ u(y  , 0) dy  dξ  n−1 (2π) Rn−1 ×Rn−1 = u(x , 0) = (γ0 u) (x )

∞ for all u ∈ C(0) (Rn+ ).

This proves that the trace operator T0 = γ0 is the Dirichlet boundary condition (see Section 5.3). Example 5.7 (Neumann condition). We let t1 (x , ξ  , τ ) = iτ     −1 n−1  − 1 ∈ S1,0 Rn−1 = i ξ   ξ   × R , ⊗π H2,τ   x ξ Then we have the formula

m = 1, d = 2.

5.3 Trace, Potential and Singular Green Operators on Rn +

1 2π 





(iτ ) R

R

197

 e−i yn τ (e+ u)(y  , yn ) dyn dτ

e−yn ξ  (e+ u)(y  , yn ) dyn       1 ∂ i τ xn −i yn τ +  = e e (e u)(y , yn ) dyn dτ  ∂xn 2π R R xn =0    ∂  + (e u)(y  , xn )  = ∂xn xn =0 

=

R

=

∂u  ∞ (y , 0) for all u ∈ C(0) (Rn+ ). ∂xn

Therefore, we have the formula for the trace operator     1  (T1 u) (x ) = ei(x −y )·ξ (2π)n−1 n−1 n−1 R ×R      1 −i yn τ +  × (iτ ) e (e u)(y , yn ) dyn dτ dy  dξ  2π R R     ∂u 1 = ei(x −y )·ξ (y  , 0) dy  dξ  (2π)n−1 ∂x n−1 n−1 n R ×R ∂u  = (x , 0) ∂xn ∞ (Rn+ ). = (γ1 u) (x ) for all u ∈ C(0) This proves that the trace operator T1 = γ1 is the Neumann boundary condition (see Section 5.3). Example 5.8 (Trace symbol and operator). We let 1 1 1 =  − iτ ξ  1 − iτ ξ  −1   −1 −  π H0,τ Rn−1 ∈ S1,0 × Rn−1 , ⊗ x ξ

t2 (x , ξ  , τ ) =

ξ  

m = −1, d = 0.

Then we have the formula for symbols 1 2π

 e R

−i yn τ

1 1 dτ = ξ   − iτ 2π



+

ei yn τ

1 dτ iτ + ξ  

= e−yn ξ  . 

This proves that   1 1 e−i yn τ (e+ u)(y  , yn ) dτ dyn 2π R ξ   − iτ R   ∞   = e−yn ξ  (e+ u)(y  , yn ) dyn = e−yn ξ  u(y  , yn ) dyn . R

0

(5.10)

198

5 Boutet de Monvel Calculus

Therefore, we have the formula for the trace operator (T2 u) (x )  ∞   1 −yn ξ   i(x −y  )·ξ   e e = u(y , yn ) dyn dy  dξ  (2π)n−1 0 Rn−1 ×Rn−1  ∞   1 −yn ξ   i x ·ξ   u 0(ξ , yn ) dyn dξ  = e e (2π)n−1 Rn−1 0 ∞ for all u ∈ C(0) (Rn+ ),

and so ∞ (Rn+ ) −→ C ∞ (Rn−1 ). T2 : C(0)

5.3.3 Singular Green Operators on Rn + A function g (x , y  , ξ  , ν, τ ) ∈ C ∞ (Rn−1 × Rn−1 × Rn−1 × R × R) is called a singular Green symbol of order m and type d if it satisfies the condition g (x , y  , ξ  , ν, τ ) ⎛ ⎞ d−1 ∞   −1 j   (1 − iν ξ  ) ⎠  −1  ⎝ τ ξ   cj (x , y  , ξ  ) = −1 (1 + iν ξ   )j+1 j=0 =0 +

∞  ∞  j=0 =0

bj (x , y  , ξ  )

−1 j

(1 − iν ξ   (1 + iν

−1 

(1 + iτ ξ  

)

−1 ξ   )j+1

(5.11)

(1 − iτ

)

. −1 ξ   )+1

Here the symbols cj (x , y  , ξ  ) form a rapidly decreasing sequence in the class m (Rn−1 × Rn−1 × Rn−1 ) with respect to j, for each  = 0, 1, 2, . . . , d − 1, S1,0 and the symbols bj (x , y  , ξ  ) form a rapidly decreasing double sequence in m the class S1,0 (Rn−1 × Rn−1 × Rn−1 ). The condition (5.11) is equivalent to the following:   − m  π Hd,τ  π Hν+ ⊗ Rn−1 g(x , y  , ξ  , ξ   ν, ξ   τ ) ∈ S1,0 × Rn−1 × Rn−1 . ⊗ x y ξ Then the singular Green operator ∞ G : C(0) (Rn+ ) −→ C ∞ (Rn+ )

is defined as an oscillatory integral by the formula (Gu)(x , xn )

(5.12)

5.3 Trace, Potential and Singular Green Operators on Rn +

199



   1 ei(x −y )·ξ n−1 (2π) Rn−1 ×Rn−1      1    −i yn τ +  × gˇ(x , y , ξ , xn , τ ) e (e u)(y , yn )dyn dτ dy  dξ  2π R R

=

∞ for all u ∈ C(0) (Rn+ ).

Here

1 gˇ(x , y , ξ , xn , τ ) = 2π 







ei xn ν g(x , y  , ξ  , ν, τ ) dν.

R

If the singular Green symbol g(x , y  , ξ  , ν, τ ) does not depend on x and  y , then the oscillatory integral (5.12) reduces to the following (see Remark 5.3): (Gu)(x , xn )    1 1 i(x ·ξ  +xn ν) = e (2π)n Rn 2π

+

+ u(ξ  , τ ) dτ g(ξ  , ν, τ ) e>



dξ  dν

∞ for all u ∈ C(0) (Rn+ ).

The mapping properties of the singular Green operator G of order m − 1 can be visualized in Figure 5.6 below. s,p Hcomp Rn +

∞ C(0) Rn +

G

s−m,p −−−−− → Hloc Rn +

−−−−− → G

C ∞ Rn +

Fig. 5.6. The condition that 1 < p < ∞ and s − 1/p ∈ Z

We give three typical examples of singular Green operators: Example 5.9 (Composition of Poisson operator and Dirichlet condition). We let g0 (x , y  , ξ  , ν, τ ) = k(x , ξ  , ν) · t0 (y  , ξ  , τ )   1 −1 n−1  −  π H1,τ Rn−1 · 1 ∈ S1,0 × R , =  ⊗π Hν+ ⊗   x ξ ξ  + iν where (see Examples 5.5 and 5.6) k(x , ξ  , ν) =

ξ  

  1 1 1 −1  π Hν+ , Rn−1 =  ∈ S1,0 × Rn−1 ⊗   x ξ −1 + iν ξ  1 + iν ξ  

200

5 Boutet de Monvel Calculus

  − 0  π H1,τ Rn−1 t0 (y  , ξ  , τ ) = 1 ∈ S1,0 × Rn−1 . ⊗ x ξ Then we have, by Examples 5.5 and 5.6, (G0 u) (x , xn ) = (K(T0 u)) (x , xn )     1 ei x ·ξ e−xn ξ  u = 0(ξ  , 0) dξ  n−1 (2π) Rn−1

∞ for all u ∈ C(0) (Rn+ ),

and so ∞ (Rn+ ) −→ C ∞ (Rn+ ). G0 = K ◦ T0 : C(0)

Here u 0(ξ  , 0) =







e−i y ·ξ u(y  , 0) dy 

Rn−1

is the partial Fourier transform of the function u(y  , 0) with respect to the variable y  . Example 5.10 (Composition of Poisson operator and Neumann condition). We let g1 (x , y  , ξ  , ν, τ ) = k(x , ξ  , ν) · t1 (y  , ξ  , τ )   1 n−1  − 0  π H2,τ · iτ ∈ S1,0 × R , =  Rn−1 ⊗π Hν+ ⊗  x ξ ξ  + iν where (see Examples 5.5 and 5.6)   1 1 1 −1 n−1 n−1  R = S × R ⊗π Hν+ ,   1,0 x ξ ξ   + iν ξ   1 + iν ξ  −1     −1 − 1  π H2,τ ∈ S1,0 Rn−1 t1 (y  , ξ  , τ ) = iτ = i ξ   τ ξ   × Rn−1 . ⊗ x ξ k(x , ξ  , ν) =

Then we have, by Examples 5.5 and 5.7, (G1 u) (x , xn ) = (K (T1 u)) (x , xn )     ∂0 u  1 ei x ·ξ e−xn ξ  (ξ , 0) dξ  = (2π)n−1 Rn−1 ∂xn

∞ for all u ∈ C(0) (Rn+ ),

and so ∞ (Rn+ ) −→ C ∞ (Rn+ ). G1 = K ◦ T1 : C(0)

Here

∂0 u  (ξ , 0 = ∂xn

 Rn−1



e−i y ·ξ



∂u  (y , 0) dy  ∂xn

is the partial Fourier transform of the function ∂u/∂xn (y  , 0) with respect to the variable y  .

5.3 Trace, Potential and Singular Green Operators on Rn +

201

Example 5.11 (Singular Green symbol and operator). We let g2 (x , y  , ξ  , ν, τ ) = k(x , ξ  , ν) · t2 (y  , ξ  , τ )   1 1 −2 −  π H0,τ  π Hν+ ⊗ Rn−1 ∈ S1,0 × Rn−1 , =  ⊗ x ξ  ξ  + iν ξ  − iτ where (see Example 5.5 and 5.8)   1 1 1 −1 n−1 n−1  R = ∈ S × R ⊗π Hν+ ,   1,0 x ξ ξ   + iν ξ   1 + iν ξ  −1   1 1 1 −1 n−1 n−1  − R =  ∈ S × R . t2 (y  , ξ  , τ ) =  ⊗π H0,τ   1,0 x ξ ξ  − iτ ξ  1 − iτ ξ  −1

k(x , ξ  , ν) =

Then we have the formula for symbols  1 1 1 dν ·  ei xn ν gˇ2 (x , y  , ξ  , xn , τ ) = 2π R iν + ξ   ξ  − iτ  + 1 1 1 dν ·  ei xn ν = 2π iν + ξ   ξ  − iτ  1 e−xn ξ  , =  ξ  − iτ and also, by formula (5.10),    1    −i yn τ +  gˇ2 (x , y , ξ , xn , τ ) e (e u)(y , yn ) dyn dτ 2π R R    1 1 −xn ξ   −i yn τ +  e (e u)(y , yn ) dyn dτ =e  R 2π ξ  − iτ R     1 1 −xn ξ   −i yn τ dτ (e+ u)(y  , yn ) dyn =e e 2π R ξ   − iτ R   −xn ξ   =e e−yn ξ  (e+ u)(y  , yn )dyn R∞  −xn ξ   =e e−yn ξ  u(y  , yn ) dyn . 0

Therefore, we have the formula for the singular Green operator (G2 u) (x , xn ) = (K (T2 u)) (x , xn )      1 = ei(x −y )·ξ e−xn ξ  (2π)n−1 Rn−1 ×Rn−1  ∞  −yn ξ    u(y , yn ) dyn dy  dξ  e × 0   ∞  1 −yn ξ   i x ·ξ  −xn ξ    = e e e u 0(ξ , yn ) dyn dξ  (2π)n−1 Rn−1 0

202

5 Boutet de Monvel Calculus ∞ for all u ∈ C(0) (Rn+ ),

and so ∞ G2 = K ◦ T2 : C(0) (Rn+ ) −→ C ∞ (Rn+ ).

5.3.4 Boundary Operators on Rn−1 Finally, we give two typical examples of boundary operators: Example 5.12 (Composition of Neumann condition and Poisson operator). We let   − 1  π H2,τ Rn−1 × Rn−1 ⊗ , t1 (y  , ξ  , τ ) = iτ ∈ S1,0 x ξ   1 1 1 −1  π Hν+ . Rn−1 =  k(x , ξ  , ν) =  ∈ S1,0 × Rn−1 ⊗   x ξ −1 ξ  + iν ξ  1 + iν ξ   Then we have the composition formula for symbols (see [19, formula (1.13), 7])  + iν 1 1 dν = − ξ   ∈ S1,0 (Rn−1 × Rn−1 ). 2π ξ   + iν Therefore, we have the composition formula for the operators T1 and K ((T1 ◦ K) v) (x ) = (T1 (Kv)) (x )    1 = ei x ·ξ (− ξ  ) v(ξ  ) dξ  (2π)n−1 Rn−1

for all v ∈ C0∞ (Rn−1 ),

and so T1 ◦ K : C0∞ (Rn−1 ) −→ C ∞ (Rn−1 ). This operator T1 ◦ K is called the Dirichlet-to-Neumann operator (see Section 6.4 in Chapter 6). Example 5.13 (Composition of potential and trace operators). We let   1 1 1 −1 n−1 n−1  R =  ∈ S × R k(x , ξ  , ν) =  ⊗π Hν+ ,   1,0 x ξ ξ  + iν ξ  1 + iν ξ  −1 and t3 (x , ξ  , τ ) = = where

q(x , ξ  ) ξ   − iτ 1

  q(x , ξ  ) n−1 n−1  − k R ∈ S × R , ⊗π H0,τ   1,0 x ξ ξ   1 − iτ ξ  −1 k+1 q(x , ξ  ) ∈ S1,0 (Rn−1 × Rn−1 ).

d = 0,

5.4 Historical Perspective of the Wiener–Hopf Technique

203

Then we have the composition formula for symbols (see [19, formula (1.13), 7])  q(x , ξ  ) 1 1 dν 2π R ξ   + iν ξ   − iν  + 1 1 1 = q(x , ξ  ) dν   2π ξ  + iν ξ  − iν   + 1 1 q(x , ξ  ) 1 + = dν 2 ξ   2π ξ   + iν ξ   − iν q(x , ξ  ) k (Rn−1 × Rn−1 ). ∈ S1,0 = 2 ξ   Therefore, we have the composition formula for the operators T3 and K ((T3 ◦ K) v) (x ) = (T3 (Kv)) (x )    1 i x ·ξ  q(x ξ ) e = v(ξ  ) dξ  (2π)n−1 Rn−1 2 ξ  

for all v ∈ C0∞ (Rn−1 ),

and so T3 ◦ K : C0∞ (Rn−1 ) −→ C ∞ (Rn−1 ).

5.4 Historical Perspective of the Wiener–Hopf Technique It should be emphasized that the Boutet de Monvel calculus is closely related to the classical Wiener–Hopf technique (see [145], [84]) that remains a source of inspiration to Mathematicians, Physicists and Engineers working in many diverse fields, and the areas of application continue to broaden. The Wiener–Hopf technique was first propounded as a means to solve, for a given function f (x), an integral equation of the form  ∞ k(x − y)f (y) dy = g(x) for 0 < x < ∞. (5.13) 0

Here: (i) k(x − y) is a known difference kernel. (ii) g(x) is a specified function defined over R+ = (0, ∞). Full details can be found in the textbook by Noble [84]. The method proceeds by extending the domain of the integral equation (5.13) to negative real values of x, that is,   ∞ g(x) for 0 < x < ∞, k(x − y)f (y) dy = (5.14) h(x) for −∞ < x < 0, 0

204

5 Boutet de Monvel Calculus

where h(x) is unknown. Then the Fourier transform of equation (5.14) yields the typical Wiener–Hopf equation G+ (α) + H− (α) = F+ (α) K(α).

(5.15)

Here: (a) F+ (α) is the half-range Fourier transform defined over the positive real axis R+ = (0, ∞) of the unknown function f (x).  ∞ F+ (α) = e−i α x f (x) dx. 0

(b) H− (α) is the half-range Fourier transform defined over the negative real axisR− = (−∞, 0) of the unknown function h(x).  H− (α) =

0

e−i α x h(x) dx.

−∞

(c) G+ (α) is the half-range Fourier transform defined over the positive real axis R+ = (0, ∞) of the known function g(x).  ∞ G+ (α) = e−i α x g(x) dx. 0

(d) K(α) is the full-range Fourier transform defined over the real axis R = (−∞, ∞) of the known function k(x). 

0

K(α) =

e−i α x k(x) dx.

−∞

We remark that the product form of the right-hand side of equation (5.15) is due to the fact that the original integral operator is of convolution type. The subscripts + and − indicate that respective functions are analytic in upper and lower half regions of the complex α-plane. The Wiener–Hopf procedure hinges on finding a product-factorization for the Fourier transformed kernel in the form K(α) = K+ (α) K− (α).

(5.16)

For example, the functions K± (α) are explicitly given by the formulas:    ∞ 1 ln K(z) K− (α) = exp − dz for Im α < 0, 2πi −∞ z − α    ∞ 1 ln K(z) dz for Im α > 0. K+ (α) = exp 2πi −∞ z − α By the factorization (5.16), we can rewrite the equation (5.15) as follows:

5.4 Historical Perspective of the Wiener–Hopf Technique

H− (α) G+ (α) + = F+ (α) K+ (α). K− (α) K− (α)

205

(5.17)

It should be noticed that the factors on the right-hand side are zero-free in their indicated half planes of analyticity. Moreover, we consider a sum-factorization of the form G+ (α) = L+ (α) + L− (α). K− (α)

(5.18)

For example, the functions L± (α) are explicitly given by the formulas:  ∞ G+ (z) 1 1 dz for Im α < 0, L− (α) = − 2πi −∞ K− (z) z − α  ∞ G+ (z) 1 1 L+ (α) = dz for Im α > 0. 2πi −∞ K− (z) z − α By using the factorization (5.18), we can express the equation (5.17) in the form H− (α) L− (α) + = F+ (α) K+ (α) − L+ (α). (5.19) K− (α) The left-hand side of the equation (5.19) is analytic in the lower half-plane, while the right-hand side of the equation (5.19) is analytic in the overlapping upper half region. Arguments involving analytic continuation enable both sides of the equation (5.19) to be equated to an entire function E(α). Moreover, physical constraints on the behavior of f (x), g(x) and k(x) as x → 0, and their Fourier transformed quantities in the equation (5.19) as |α| → ∞ allows us to specify the function E(α): E(α) = L− (α) +

H− (α) = F+ (α) K+ (α) − L+ (α). K− (α)

In this way, the functions H− (α) and F+ (α) are (uniquely) determined respectively as follows: H− (α) = (E(α) − L− (α)) K− (α), F+ (α) =

E(α) + L+ (α) . K+ (α)

Finally, by using the Fourier inversion formula we can determine the unknown function    ∞ E(α) + L+ (α) E(α) + L+ (α) 1 f (x) = F ∗ dα. ei x α = K+ (α) 2π −∞ K+ (α)

206

5 Boutet de Monvel Calculus

5.5 Notes and Comments The material of this chapter is adapted from Boutet de Monvel [17], [18], [19], Noble [84], Rempel–Schulze [93], Schrohe [101] and also Taira [122, Chapter 7, Section 7.7]. Section 5.3: This section is devoted to several important examples of Boutet de Monvel calculus with the special emphasis on elliptic boundary value problems in terms of pseudo-differential operators. These examples are taken from Taira [124] and [126].

6 Lp Theory of Elliptic Boundary Value Problems

In this chapter we consider the non-homogeneous general Robin problem ⎧ in Ω, ⎨Au = f   (6.1) ∂u ⎩Bγu = a(x ) + b(x )u = ϕ on ∂Ω ∂ν ∂Ω

under the following two conditions (H.1) and (H.2) (corresponding to conditions (A) and (B) with μ = a and γ = −b): (H.1) a(x ) ≥ 0 and b(x ) ≥ 0 on ∂Ω. (H.2) a(x ) + b(x ) > 0 on ∂Ω. Here ν = −n is the unit outward normal to the boundary ∂Ω (see Figure 6.1 below). The general Robin boundary operator Bγ is defined as follows: Bγ = a(x ) γ1 + b(x ) γ0 ,

(6.2)

where the trace map γ = (γ0 , γ1 ) is given by the formulas  γ0 u = u|∂Ω ,   ∂u  ∂u  = − ∂n . γ1 u = ∂ν ∂Ω ∂Ω Section 6.1 is devoted to the study of the classical surface and volume potentials arising in boundary value problems for elliptic differential operators, in terms of pseudo-differential operators, This calculus of pseudo-differential operators is applied to elliptic boundary value problems in Part III. In Section 6.2 we consider the Dirichlet problem in the framework of Sobolev spaces of Lp type. This is a generalization of the classical potential approach to the Dirichlet problem.

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 6

208

6 Lp Theory of Elliptic Boundary Value Problems ν = −n ∂Ω

Ω

Fig. 6.1. The bounded domain Ω and the outward normal ν to the boundary ∂Ω

In Section 6.3 we formulate elliptic boundary value problems in the framework of Sobolev spaces of Lp type. The pseudo-differential operator approach to elliptic boundary value problems can be traced back to the pioneering work of Calder´ on [22] in early 1960s ([57], [105]). In Section 6.4, by using the Poisson operator P and the Green operator GN for the Neumann problem we show that the general Robin problem (6.1) can be reduced to the study of a pseudo-differential operator T = BγP = a(x ) γ1 P + b(x ) γ0 P = −a(x ) Π + b(x ) on the boundary, where Π is the Dirichlet-to-Neumann operator given by the formula   ∂ (Pϕ) for all ϕ ∈ C ∞ (∂Ω). (6.3) Πϕ := −γ1 (Pϕ) = ∂n ∂Ω

The virtue of this reduction to the boundary is that there is no difficulty in taking adjoints or transposes after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints or transposes. This allows us to discuss the existence theory more easily. In Section 6.5 we study the non-homogeneous general Robin problem (6.1) and the non-homogeneous Neumann problem ⎧ in Ω, ⎨Av =  g (6.4) ∂v  ⎩ = ψ on ∂Ω ∂n ∂D in the framework of Lp Sobolev spaces from the viewpoint of the Boutet de Monvel calculus. Then we derive an index formula of Agranoviˇc–Dynin type for the Neumann problem (6.4) and the general Robin problem (6.1) in the framework of Lp Sobolev spaces (Theorem 6.26). In Section 6.6 we study an intimate relationship between the Dirichlet-toNeumann operator Π and the reflecting diffusion in a bounded, domain Ω of Euclidean space RN with smooth boundary ∂D (Theorem 6.27). This section is a probabilistic approach to pseudo-differential operators. In Section 6.7, following Mizohata [82, Chapter 3] and Wloka [146] we prove that all the sufficiently large eigenvalues of the Dirichlet problem for the differential operator A lie in the parabolic type region (see assertion (6.74)).

6.1 Classical Potentials and Pseudo-Differential Operators

209

Hence, by considering A−λ1 and A∗ −λ1 we may assume that the fundamental condition (SC) N0 (A − λ1 ) = N0 (A∗ − λ1 ) = {0} is satisfied for certain positive number λ1 .

6.1 Classical Potentials and Pseudo-Differential Operators The purpose of this section is to describe, in terms of pseudo-differential operators, the classical surface and volume potentials arising in boundary value problems for elliptic differential operators. This calculus of pseudo-differential operators will be applied to elliptic boundary value problems in Chapters 12, 13 and 14. 6.1.1 Single and Double Layer Potentials Let Δ be the usual Laplacian Δ=

∂2 ∂2 ∂2 + + . . . + . ∂x21 ∂x22 ∂x2n

Then the Newtonian potential N (x − y) is given by the formula (−Δ)−1 f (x) = N ∗ f (x)

 1 Γ ((n − 2)/2) f (y) dy = |x − y|n−2 4π n/2 n  R 1 1 f (y) dy for all f ∈ C0∞ (Rn ), = (n − 2)ωn Rn |x − y|n−2

where ωn =

2π n/2 Γ (n/2)

is the surface area of the unit sphere. In the case n = 3, we have the formula  f (y) 1 dy. u(x) = 4π R3 |x − y| Up to an appropriate constant of proportionality, the Newtonian potential N (x − y) is the gravitational potential at position x due to a unit point mass at position y, and so the function u(x) is the gravitational potential due to a continuous mass distribution with density f (x). In terms of electrostatics, the function u(x) describes the electrostatic potential due to a charge distribution with density f (x). We define a single layer potential with density ϕ by the formula

210

6 Lp Theory of Elliptic Boundary Value Problems

N ∗ (ϕ(x ) ⊗ δ(xn ))  Γ ((n − 2)/2) ϕ(y  ) = dy  n/2   2 2 )(n−2)/2 4π (|x − y | + x n−1 R n  1 ϕ(y  ) = dy  for all ϕ ∈ C0∞ (Rn−1 ). (n − 2)ωn Rn−1 (|x − y  |2 + x2n )(n−2)/2 In the case n = 3, the function N ∗ (ϕ ⊗ δ) is related to the distribution of electric charge on a conductor Ω. In equilibrium, mutual repulsion causes all the charge to reside on the surface ∂Ω of the conducting body with density ϕ, and ∂Ω is an equipotential surface. We define a double layer potential with density ψ by the formula  xn ψ(y  ) 1   dy  N ∗ (ψ(x ) ⊗ δ (xn )) = ωn Rn−1 (|x − y  |2 + x2n )(n−2)/2 for all ψ ∈ C0∞ (Rn−1 ). In the case n = 3, the function N ∗ (ψ ⊗ δ  ) is the potential induced by a distribution of dipoles on R2 with density ψ(y  ), the axes of the dipoles being normal to R2 . On the other hand, it is easy to verify that if ϕ is bounded and continuous on Rn−1 , then the function  xn 2 u(x , xn ) = ϕ(y  ) dy  (6.5)   ωn Rn−1 (|x − y |2 + x2n )n/2 is well-defined for (x , xn ) ∈ Rn+ , and is a (unique) solution of the Dirichlet problem  Δu = 0 in Rn+ , γ0 u = u|Rn−1 = ϕ on Rn−1 . Formula (6.5) is called the Poisson integral formula for the solution of the Dirichlet problem. Furthermore, by using the Fourier transform we can express formula (6.5) for ϕ ∈ S(Rn−1 ) as follows:     1 u(x , xn ) = ei x ·ξ e−xn |ξ | ϕ(ξ   ) dξ  . (6.6) n−1 (2π) n−1 R To do this, we need the following elementary formulas: Claim 6.1. (1) For any a > 0, we have the formula 

∞

2

ei α x e−a x dx =

−∞

(2) For any β > 0, we have the formula

8

α2 π − e 4a . a

(6.7)

6.1 Classical Potentials and Pseudo-Differential Operators

e

−β

1 = √ π



∞

0

β2 e−s − √ e 4s ds. s

211

(6.8)

Proof. We only prove formula (6.8). To do so, we remark that the function C  z −→ f (z) =

ei β z 1 + z2

has a pole at z = i in the closed half-plane {z ∈ C : Im z ≥ 0}, and further that its residue is given by the formula Res [f (z)]z=i = lim (z − i)f (z) = z→i

i e−β = − e−β . 2i 2

Hence we have, by the residue theorem,    i f (z) dz = 2πi − e−β = π e−β . 2 Γ

(6.9)

Here Γ is a path consisting of the semicircle and the segment as in Figure 6.2 below.

iR

Γ

C

−R

0

R

Fig. 6.2. The integral path Γ consisting of the semicircle C completed by the segment [−R, R]

Then we can rewrite formula (6.9) as follows:    R −β f (z) dz = f (z) dz + f (x) dx := I + II. πe = Γ

CR

−R

However, since we have, for all z = x + iy ∈ C,  i β z   i β x −β y   −β y  e  = e  = e  ≤ 1, e we can estimate the first term I as follows:     π iθ     ei β R e  iθ   f (z) dz R i e dθ =      2 2iθ  0 1+R e CR

(6.10)

212

6 Lp Theory of Elliptic Boundary Value Problems

 ≤

0

π

πR R dθ = 2 −→ 0 −1 R −1

R2

as R → ∞.

Therefore, by using Fubini’s theorem we obtain from formula (6.7) with α := β and a := s that  R  1 1 ∞ ei β x lim f (x) dx = dx π R→∞ −R π −∞ 1 + x2   ∞  2 1 ∞ iβ x = e e−(1+x )s ds dx π −∞ 0   ∞  ∞ 2 1 = e−s ei β x e−s x dx ds π 0 −∞  ∞ −s β2 1 e √ e− 4s ds. =√ π 0 s

e−β =

This proves the desired formula (6.8). The proof of Claim 6.1 is complete.   Therefore, it follows from an application of Fubini’s theorem and Claim 6.1 with β := xn |ξ  | that     1 ei x ·ξ e−xn |ξ | ϕ(ξ   ) dξ  n−1 (2π) Rn−1     1 i(x −y  )·ξ  −xn |ξ  |  = ϕ(y  ) e e dξ dy  (2π)n−1 Rn−1 Rn−1     ∞ 1 1 −s(1+|x −y  |2 /x2n ) n/2−1 s = ϕ(y  ) e ds dy  π n/2 xn−1 n 0 Rn−1     ∞ 1 1 xnn −s n/2−1 = ϕ(y  ) e s ds dy  π n/2 xn−1 (|x − y  |2 + x2n )n/2 n 0 Rn−1  xn Γ (n/2) = ϕ(y  ) dy  n/2   2 2 n/2 π Rn−1 (|x − y | + xn )  xn 2 = ϕ(y  ) dy  .   ωn Rn−1 (|x − y |2 + x2n )n/2 This proves the desired formula (6.6).   6.1.2 The Green Representation Formula First, we have the jump formula (4.15) for the minus Laplacian −Δ in the half-space Rn+ :   ∂u  0 (x , 0) ⊗ δ(xn ) − u(x , 0) ⊗ δ  (xn ). (−Δ) u0 = (−Δu) − ∂xn

6.1 Classical Potentials and Pseudo-Differential Operators

213

By applying the Newtonian potential N (x − y) to the both sides (see Subsection 6.1.1), we obtain that   u0 = (N ∗ (−Δ)) u0   = N ∗ (−Δu)0 − N ∗ (γ1 u ⊗ δ(xn )) − N ∗ (γ0 u ⊗ δ  (xn ))   ∂u  N (x − y) Δu(y) dy − N (x − y  , xn ) (y , 0) dy  =− ∂y n Rn Rn−1  ∂N  + (x − y  , xn ) u(y  , 0) dy. ∂y n−1 n R Therefore, we arrive at the classical Green representation formula:  1 1 u(x) = Δu(y) dy n (2 − n)ωn R+ |x − y|n−2  1 ∂u  1 (y , 0) dy  + (2 − n)ωn Rn−1 (|x − y  |2 + x2n )(n−2)/2 ∂yn  1 xn + u(y  , 0) dy  for x ∈ Rn+ .   ωn Rn−1 (|x − y |2 + x2n )n/2 We remark that the first term is the Newtonian potential and the second and third terms are the single and double layer potentials, respectively. 6.1.3 Surface and Volume Potentials We give a formal description of a background. Let Ω be a bounded domain in Euclidean space Rn with smooth boundary ∂Ω. Its closure Ω is an ndimensional, compact smooth manifold with boundary. We may assume that Ω is the closure of a relatively compact, open subset Ω of an n-dimensional compact smooth manifold M without boundary in which Ω has a smooth boundary ∂Ω (see Figure 4.1). The manifold M is called the double of Ω. Let P be a differential operator of order m with smooth coefficients on M . Then we have the jump formula (4.15)   0 P u0 = (P u) + P0γu

for all u ∈ C ∞ (Ω),

(6.11)

where u0 is the extension of u to M by zero outside Ω, and P0γu is a distribution on M with support in ∂Ω. If P admits an “inverse” Q, then the function u may be expressed as follows:      0  u = Q (P u)  + Q P0 γu  . Ω

Ω

The first term on the right-hand side is a volume potential and the second term is a surface potential with m “layers”. For example, if P is the usual Laplacian Δ and if Ω = Rn+ , then the first term is the classical Newtonian

214

6 Lp Theory of Elliptic Boundary Value Problems

potential and the second term is the familiar combination of single and double layer potentials (see Subsection 6.1.2). (I) First, we state a theorem which covers surface potentials (see [17, Th´eor`eme (1.3.5)], [26, Chapitre V, Th´eor`eme 2.4]): Theorem 6.2. Let A ∈ Lm cl (M ) be properly supported. Assume that &∞ Every term in the complete symbol j=0 aj (x, ξ) of A

(6.12)

is a rational function of ξ Then we have the following three assertions (i), (ii) and (iii): (i) The operator H : v −→ A(v ⊗ δ)|Ω ∞

is continuous on C (∂Ω) into C ∞ (Ω). If v ∈ D (∂Ω), the distribution Hv has sectional traces on ∂Ω of any order. (ii) The operator S : C ∞ (∂Ω) −→ C ∞ (∂Ω) v −→ Hv|∂Ω belongs to the class Lm+1 (∂Ω). Furthermore, its homogeneous principal cl symbol is given by the formula  1 (x , ξ  ) −→ a0 (x , 0, ξ  , ξn ) dξn (6.13) 2π Γ where a0 (x , xn , ξ  , ξn ) ∈ C ∞ (T ∗ (M ) \ {0}) is the homogeneous principal symbol of A, and Γ is a circle in the plane {ξn ∈ C : Im ξn > 0} that encloses the poles ξn of a0 (x , 0, ξ  , ξn ) there (see Figure 6.3 below). (iii) If 1 < p < ∞, then the operator H extends to a continuous linear operator H : B s,p (∂Ω) −→ H s−m−1+1/p,p (Ω) for all s ∈ R (see Figure 6.4 below). Proof. We only prove the case where p = 2, by following [26, Chapitre V, Th´eor`eme 2.4]. By using local coordinate systems flattening out the boundary ∂Ω, together with a partition of unity, we may assume that Ω = Rn+ , u ∈ E  (Rn ). The proof of Theorem 6.2 is divided into three parts. Proof of Assertion (i): As in the proof of Theorem 4.19, we have, for all Φ(x) ∈ C0∞ (Rn ),

6.1 Classical Potentials and Pseudo-Differential Operators

215

Γ

Fig. 6.3. The circle Γ in the plane {Im ξn > 0} that encloses the poles ξn of a0 (x , 0, ξ  , ξn ) H

B s,p (∂Ω) −−−−− → H s−m−1+1/p,p (∂Ω)

C ∞ (∂Ω) −−−−− → H

C ∞ (Ω)

Fig. 6.4. The mapping properties of the surface potential H for all s ∈ R and 1 0 is a constant. On the other hand, since μ < −1, it follows that  μ /2  μ /2 1 + |ξ  |2 + ξn2 ≤ 1 + ξn2 for all ξ = (ξ  , ξn ) ∈ Rn . Hence we have, for all x, y ∈ Rn ,  i x·ξ   e a (x, ξ) v(ξ  ) − ei y·ξ a (y, ξ) v(ξ  ) μ /2  | v (ξ  )| ∈ L1 (Rn ). ≤ C 1 + ξn2 By applying Lebesgue’s dominated convergence theorem, we find that A (v(x ) ⊗ δ(xn )) ∈ C(Rn ). The proof of Claim 6.4 is complete.  

218

6 Lp Theory of Elliptic Boundary Value Problems

Step (2): By Claim 6.4, it follows that    1   ei x ·ξ a (x , 0, ξ)  v (ξ  ) dξ ∈ C(Rn−1 ). A (v ⊗ δ) (x , 0) = (2π)n Rn Moreover, since μ < −1, we can write the formula A (v ⊗ δ) (x , 0) in the form A (v ⊗ δ) (x , 0)     ∞ 1 1 i x ·ξ     = e a (x , 0, ξ , ξn ) dξn v(ξ  ) dξ  . (2π)n−1 Rn−1 2π −∞ Indeed, it suffices to note that  μ /2 |a (x , 0, ξ  , ξn )| ≤ C 1 + |ξ  |2 + ξn2 μ /2  ≤ C 1 + ξn2 for all ξn ∈ R. More precisely, we have the following claim: Claim 6.5. The operator C0∞ (Rn−1 )  v −→ A (v ⊗ δ) (x , 0) is a pseudo-differential operator on Rn−1 with symbol  ∞    1    a (x , ξ ) := a (x , 0, ξ  , ξn ) dξn ∈ S μ +1 Rn−1 × Rn−1 . 2π −∞ Proof. For each compact set K  ⊂ Rn−1 and multi-indices α , β  , there exists a constant Cα ,β  ,K  > 0 such that      μ /2−|β  |/2   α β   Dx Dξ a (x , 0, ξ  , ξn ) ≤ Cα ,β  ,K  1 + |ξ  |2 + ξn2 for all x ∈ K  and ξ = (ξ  , ξn ) ∈ Rn . Hence we have the inequality      α β     Dx Dξ a (x , ξ )    ∞   1  α β     Dx Dξ a (x , 0, ξ , ξn ) dξn  = 2π −∞  μ /2−|β  |/2 Cα ,β  ,K  ∞  ≤ 1 + |ξ  |2 + ξn2 dξn 2π −∞  μ /2−|β  |/2+1/2 ∞  μ /2−|β  |/2 Cα ,β  ,K   = dηn 1 + ηn2 1 + |ξ  |2 2π −∞  (μ +1)/2−|β  |/2 ∞  μ /2 Cα ,β  ,K   1 + |ξ  |2 1 + ηn2 ≤ dηn 2π −∞  (μ +1)/2−|β  |/2 := Cα  ,β  ,K  1 + |ξ  |2 for all x ∈ K  and ξ  ∈ Rn−1 . The proof of Claim 6.5 is complete.  

6.1 Classical Potentials and Pseudo-Differential Operators

219



Summing up, we have proved that if A ∈ Lμcl (Rn ) with μ < −1, there  exists a pseudo-differential operator A ∈ Lμcl +1 (Rn−1 ) such that A (v ⊗ δ) (x , 0) = A v(x ) for all v ∈ C0∞ (Rn−1 ). Step (3): In this way, we are reduced to the study of the operator B (v ⊗ δ) (x , 0)    = ei x ·ξ k(x , 0, ξ  ) v(ξ  ) dξ  Rn−1

where 





k(x , 0, ξ ) =

for all v ∈ C0∞ (Rn−1 ),

a(x , 0, ξ  , ξn ) dξn .

Γξ

However, we can prove the following claim: Claim 6.6. The function k(x , 0, ξ  ) is positively homogeneous of degree μ + 1 with respect to the variable ξ  , for |ξ  ≥ 1. In particular, we have the assertion  μ+1  n−1 R k(x , 0, ξ  ) ∈ Scl × Rn−1 . Proof. Since we have the formula Γξ = |ξ  | Γ1

for all |ξ  ≥ 1,

it follows that 





k(x , 0, ξ ) =





a(x , 0, ξ , ξn ) dξn = Γξ

 |ξ  | Γ1





a (x , 0, ξ , ξn ) |ξ | d

  ξ |ξ  |μ a x , 0,  , ηn |ξ  | dηn |ξ | Γ1    ξ μ+1 = |ξ  | a x , 0,  , ηn dηn |ξ | Γ1    ξ μ+1 = |ξ  | k x , 0,  for all |ξ  ≥ 1. |ξ | 





ξn |ξ  |



=

The proof of Claim 6.6 is complete.   Proof of Assertion (iii): The proof is divided into six steps. Step (1): For every function ϕ ∈ C0∞ (Rn ), we have only to show that the mapping ϕK : H s (Rn−1 ) −→ H s−μ−1/2 (Rn+ ) (6.17) is continuous for all s ∈ R. Indeed, since A is properly supported, we can find a function ψ ∈ C0∞ (Rn−1 ) such that

6 Lp Theory of Elliptic Boundary Value Problems

220

for all v ∈ D (Rn−1 ).

ϕ Kv = ϕ K(ψv) Then we have the assertion vj −→ 0

s in Hloc (Rn−1 )

=⇒ ψ vj −→ 0 in H s (Rn−1 ) =⇒ ϕ Kvj = ϕ K(ψ vj ) −→ 0

in H s−μ−1/2 (Rn+ ).

This proves that s−μ−1/2

Kvj −→ 0

in Hloc

(Rn+ ).

Step (2): Now we assume that 

A ∈ Lμcl (Rn ) and that

for μ < μ,

s − μ + μ < 0.

Then we have, for all ϕ ∈ C0∞ (Rn ) and v ∈ C0∞ (Rn−1 ),

ϕ A (v(x ) ⊗ δ(xn ))|Ω s−μ−1/2 ≤ C v(x ) ⊗ δ(xn ) s−μ−1/2+μ ≤ C  v s−(μ−μ ) ≤ C  v s . Indeed, it suffices to note that 2

v(x ) ⊗ δ(xn ) s−μ−1/2+μ   s−μ−1/2+μ 1 2 1 + |ξ  |2 + |ξn |2 = | v (ξ  )| dξ  dξn n (2π) Rn  ∞    1 2 s−μ−1/2+μ 1 + |η = | dηn n (2π)n −∞   s−μ+μ 2 × | v (ξ  )| dξ  1 + |ξ  |2 Rn−1

2

= Cs,μ,μ v s−(μ−μ )

for 2(s − μ + μ − 1/2) < −1.

However, we have the decomposition ϕ A (v ⊗ δ) = ϕ B (v ⊗ δ) + ϕ R (v ⊗ δ) , where A = B + R,

B ∈ Lμcl (Rn ), R ∈ L−∞ (Rn ).

By applying inequality (6.18) to the operator R, we obtain that

ϕ R (v ⊗ δ)|Ω s−μ−1/2 ≤ Cs v s .

(6.18)

6.1 Classical Potentials and Pseudo-Differential Operators

221

Therefore, we are reduced to the proof of the following inequality:

ϕ B (v ⊗ δ)|Ω s−μ−1/2 ≤ Cs v s

for all v ∈ C0∞ (Rn−1 ).

(6.19)

From now on, we may assume that 1 = (2π)n−1









• B (v(x ) ⊗ δ(xn ))|Rn ei x ·ξ k(x, ξ  ) v(ξ  ) dξ  , + Rn−1   • k(x, ξ ) = ei xn ξn a(x, ξ  , ξn ) dξn , Γξ

where the support of a(x, ξ) (and hence that of k(x, ξ  )) with respect to x is compact in Rn+ , depending on ϕ ∈ C0∞ (Rn ). Step (3): First, we prove the following lemma: Lemma 6.7. For any multi-indices α and β, there exists a constant Cα,β > 0 such that  β α  x D k(x, ξ  ) ≤ Cα,β (1 + |ξ  |)μ+1+αn −βn (6.20) x for all x = (x , xn ) ∈ Rn+ and ξ  ∈ Rn−1 . Here the support of k(x, ξ  ) with respect to x is compact in Rn+ . Proof. A general term xβ Dxα k(x, ξ  ) is expressed as follows:    β x xβnn ei xn ξn ξnp Dxα a(x, ξ  , ξn ) dξn for 0 ≤ p ≤ αn . Γξ

However, we remark that  β ix ξ  xnn e n n  = (xn |ξ  |)βn e−(xn |ξ |) Im ζn



1 |ξ  | for all xn ≥ 0 and ζn ∈ Γ1 ,

and further that dξn = |ξ  | dζn

for ξn ∈ Γξ and ζn ∈ Γ1 .

Therefore, we have the desired assertion  β α    x Dx k(x, ξ  ) = O |ξ  |−βn +αn +μ+1 , since the support of a(x, ξ) is compact with respect to x. The proof of Lemma 6.7 is complete.  

βn

222

6 Lp Theory of Elliptic Boundary Value Problems

Step (4) We extend the function k(x, ξ  ) to the whole space Rn with respect to x in the following way: By virtue of Seeley [103] (see [123, Lemma 4.22]), we can find a function w(t) ∈ S(R) such that supp w ⊂ [1, ∞),  ∞ tn w(t) dt = (−1)n

for each non-negative integer n.

0

Moreover, we take a function φ(t) ∈ C0∞ (R) such that supp φ ⊂ [−2, 2], φ(t) = 1 If we let

 



k(x , xn , ξ ) :=

on the interval [−1, 1].

k(x , xn , ξ  ) +∞   0 w(s) φ (−xn s) k(x , −s xn , ξ ) ds

for xn ≥ 0, for xn < 0,

then we have the following three assertions (a), (b) and (c): (a) k(x , xn , ξ  ) is smooth with respect to the variable xn , in the interval (−∞, 0). (b) The support of k(x, ξ  ) with respect to x is compact in the whole space Rn . (c) k(x, ξ  ) satisfies the same estimate (6.20) for all x = (x , xn ) ∈ Rn and ξ  ∈ Rn−1 . Step (5): Moreover, we can prove the following lemma: Lemma 6.8. Let  k(ζ, ξ  ) =



e−i x·ζ k(x, ξ  ) dx

Rn

for ζ ∈ Rn and ξ  ∈ Rn−1 .

For any integers q ≥ n and r ≥ 1, there exists a constant C = Cq,r > 0 such that  −q   |ζn |     μ  −r (6.21) 1+ k(ζ, ξ ) ≤ C (1 + |ξ |) (1 + |ζ |) 1 + |ξ  | for all ζ ∈ Rn and ξ  ∈ Rn−1 . Proof. For any multi-index α, it follows from an application of Lemma 6.7 that there exists a constant Cα,q > 0 such that (1 + |x |) (1 + |xn | (1 + |ξ  |)) q



μ+1+αn

≤ Cα,q (1 + |ξ |)

−2

|Dxα k(x, ξ  )| 

for all x = (x , xn ) ∈

(6.22) Rn+



and ξ ∈ R

n−1

.

6.1 Classical Potentials and Pseudo-Differential Operators

223

We recall that the support of k(x, ξ  ) with respect to x is compact in the whole space Rn . Hence, by integration by parts it follows from inequality (6.22) that       −i x·ζ   α   α    (6.23) Dx e k(x, ξ ) ζ k(ζ, ξ ) =  n R    =  e−i x·ζ Dxα (k(x, ξ  )) Rn  ∞ 1  μ+1+αn ≤ C1 (1 + |ξ |) 2 dxn −∞ (1 + |xn | (1 + |ξ  |))  1 × dx  q Rn−1 (1 + |x |) for all ζ ∈ Rn and ξ  ∈ Rn−1 . However, we have the formula  ∞ dxn −∞

 2

(1 + |xn | (1 + |ξ  |))

∞

= −∞

=

dyn (1 + |yn |)

2

(1 + |ξ  |)

−1

2 . 1 + |ξ  |

Hence we have, by inequality (6.23),   μ+αn  α  . ζ k(ζ, ξ  ) ≤ C2 (1 + |ξ  |) Therefore, by taking α = (α , q),

|α | = r,

we obtain the desired inequality  q   |ζn | r μ    (1 + |ζ  |) 1 + ) k(ζ, ξ  ≤ C3 (1 + |ξ  |) .  1 + |ξ  | The proof of Lemma 6.8 is complete.   Step (6): If we let 1 U (x) := (2π)n





Rn−1



ei x ·ξ k(x, ξ  ) v(ξ  )dξ 

for x ∈ Rn ,

then we have the formula U |Rn = B (v ⊗ δ)|Rn . +

+

Hence it suffices to show that there exists a constant Cs > 0 such that

224

6 Lp Theory of Elliptic Boundary Value Problems

U s−μ−1/2 ≤ Cs v s ,

v ∈ C0∞ (Rn−1 ),

or equivalently |U, ϕ| ≤ Cs v s ϕ −s+μ+1/2 Here recall that U, ϕ =

1 (2π)n

 Rn

for all ϕ ∈ C0∞ (Rn ).

(6.24)

 (η) ϕ(−η) U  dη

is the pairing between the spaces H s−μ−1/2 (Rn ) and H −s+μ+1/2 (Rn ).  (η): Substep (6-a): First, we calculate the Fourier transform U   (η) = e−i x·η U (x) dx U n R    1 −i x·η i x ·ξ     = e e k(x, ξ ) v  (ξ ) dξ dx. (2π)n Rn Rn−1 However, we remark that the support of k(x, ξ  ) is compact with respect to x and that v ∈ S(Rn−1 ). Hence, by using Fubini’s theorem we have the formula    −i x ·η  −ixn ηn   (η) = 1 e U k(x, ξ ) dx v(ξ  ) dξ  (2π)n Rn−1  1  = k(η − (ξ  , 0), ξ  ) v(ξ  ) dξ  (2π)n Rn−1  1  = k(η  − ξ  , ηn , ξ  ) v(ξ  ) dξ  . (2π)n Rn−1 Therefore, we have the formula U, ϕ  1  (η) ϕ(−η) U  dη = (2π)n Rn   1      = k(η − ξ , ηn , ξ ) v(ξ ) dξ ϕ(−η)  dη (2π)n Rn−1 (−s+μ+1/2)/2 1  ϕ(−η)  = 1 + |η|2 (2π)n      1   2 (s−μ−1/2)/2    × , η , ξ ) 1 + |η| v  (ξ ) dξ k(η − ξ dη. n (2π)n−1 Rn−1 If we let



V (η) := Rn−1

     s−μ−1/2 | v (ξ  )| dξ  , k(η − ξ  , ηn , ξ  ) (1 + |η|)

by applying Schwarz’s inequality we can find a constant C > 0 such that

6.1 Classical Potentials and Pseudo-Differential Operators

|U, ϕ| ≤ C ϕ −s+μ+1/2 V 0 .

225

(6.25)

We are reduced to the study of the function V (η). Substep (6-b): The estimate of the norm V |0 . By Lemma 6.8, there exists a constant C = Cq,r > 0 such that       μ −r 1+ k(η − ξ  , ηn , ξ  ) ≤ C (1 + |ξ  |) (1 + |η  − ξ  |)

|ηn | 1 + |ξ  |

−q .

Hence we have the inequality  V (η) ≤ Cq,r

−r

(1 + |ξ  |)μ (1 + |η  − ξ  |)

 1+

Rn−1 s−μ−1/2

|ηn | 1 + |ξ  |

−q

× (1 + |η|) | v (ξ  )| dξ   = Cq,r (1 + |ξ  |)s | v (ξ  )| χ(η, ξ  )dξ  , Rn−1

where −r

χ(η, ξ  ) := (1 + |ξ  |) (1 + |η|) (1 + |η  − ξ  |)  −q |ηn | × 1+ for η = (η  , ηn ) ∈ Rn and ξ  ∈ Rn−1 . 1 + |ξ  | μ−s

If we let



s−μ−1/2

χ(η, ξ  ) (1 + |ξ  |) | v (ξ  )| dξ  s

W (η) := Rn−1

for η ∈ Rn ,

then we have the inequality V (η) ≤ Cq,r W (η)

for all η ∈ Rn ,

and so

V 0 ≤ Cq,r W 0 .

(6.26)

In this way, we are reduced to the study of the function W (η). Substep (6-c): The estimate of the norm W |0 . To do this, we make the change of variables η  = ζ  , ηn = (1 + |ζ  |) ζn . Then we have the formula   2 |W (η  , ηn )| dη  dηn = Rn

Rn

However, we have the following: (1) 1 + |η| = (1 + |ζ  |) (1 + |ζn |).

2

|W (ζ  , (1 + |ζ  |) ζn )| (1 + |ζ  |) dζ  dζn .

226

6 Lp Theory of Elliptic Boundary Value Problems

(2) 1 + |ξ  | ≤ 1 + |ζ  | + |ξ  − ζ  | ≤ (1 + |ξ  |) (1 + |ξ  − ζ  |), so that |ζn | (1 + |ζ  |)|ηn | |ηn | ≤ = .    1 + |ζ − ξ | 1 + |ξ | 1 + |ξ  | On the other hand, by applying Peetre’s inequality (see [26, p. 79, Lemme 2.6.2]) σ  1 + |ξ  |2  |σ|  σ 1 + |ζ  |2 ≤ 2|σ| 1 + |ξ  − ζ  |2 for all ξ  , ζ  ∈ Rn−1 and σ ∈ R, we have the inequality (1 + |ξ  |)

μ−s

≤ 2|μ−s|/2 (1 + |ξ  − ζ  |)

|μ−s|

(1 + |ζ  |)

μ−s

.

Summing up, we obtain that χ(ζ  , (1 + |ζ  |) ζn , ξ  ) (1 + |ζ  |) = (1 + |ξ  |)  −q |ζn | × 1+ 1 + |ζ  − ξ  | μ−s

s−μ−1/2

(1 + |ζn |)

≤ 2|μ−s|/2 (1 + |ξ  − ζ  |)

|μ−s|

= 2|μ−s|/2 (1 + |ζn |)   −1/2 × (1 + |ζ |) 1+

(1 + |ξ  − ζ  |) −q |ζn | . 1 + |ζ  − ξ  |

s−μ−1/2

(1 + |ζ  |)

(1 + |ζ  − ξ  |)

−r

(1 + |ζ  |)  −q |ζn | s−μ−1/2   −r × (1 + |ζn |) (1 + |ζ − ξ |) 1+ 1 + |ζ  − ξ  | s−μ−1/2

μ−s

s−μ−1/2

|μ−s|−r

Therefore, we have the inequality  |W (η)|2 dη n R 2 = |W (ζ  , (1 + |ζ  |) ζn )| (1 + |ζ  |) dζ  dζn Rn





χ(ζ  , (1 + |ξ  |) ζn , ξ  ) (1 + |ξ  |) | v (ξ  )| dξ  s

= Rn

(6.27)

Rn−1 

2

× (1 + |ζ |) dζ  dζn   |μ−s|−r ≤ 2|μ−s| (1 + |ζn |)s−μ−1/2 (1 + |ζ  − ξ  |) Rn 

Rn−1

−1/2

× (1 + |ζ |)

 1+

|ζn | 1 + |ζ  − ξ  |

−q



s



(1 + |ξ |) | v (ξ )| dξ



2

6.1 Classical Potentials and Pseudo-Differential Operators

227

× (1 + |ζ  |) dζ  dζn   s−μ−1/2 1 + |ξn |2 ≤C Rn

 ×



Rn−1

|μ−s|−r



(1 + |ζ − ξ |)

 1+

|ζn | 1 + |ζ  − ξ  |

−q

2 s × (1 + |ξ  |) | v (ξ  )| dξ  dζ  dζn   s−μ−1/2 1 + |ξn |2 =C n R |μ−s|−r+q −q × (1 + |ζ  − ξ  |) (1 + |ζ  − ξ  | + |ζn |) Rn−1

× (1 + |ξ  |) | v (ξ  )| dξ  s

If we let

2



• Y (ζ  , ζn ) :=

dζ  dζn .

Z(ζ  − ξ  , ζn ) (1 + |ξ  |) | v (ξ  )| dξ  , s

Rn−1

|μ−s|−r+q

• Z(ζ  , ςn ) := (1 + |ζ  |)

(1 + |ζ| + |ζn |)−q ,

then we can rewrite inequality (6.27) as follows:   s−μ−1/2 2 2

W 0 ≤ C 1 + |ξn |2

Y (·, ζn 0 dζn .

(6.28)

R

However, by the Hausdorff–Young inequality it follows that 

Y (·, ζn 0 ≤



n−1

R 

≤C

|Z(ζ , ζn )| dζ



  

Rn−1



Rn−1

2s

|Z(ζ  , ζn )| dζ  v s .

q > n − 1, r ≥ |μ − s| + q, then we have the inequality

Rn−1







|Z(ζ , ζn )| dζ = ≤

R

|μ−s|−r+q

n−1

Rn−1

2

(1 + |ξ |) | v (ξ )| dξ

Moreover, if we choose q and r so large that





(1 + |ζ  |)  q dζ (1 + |ζ  | + |ζn |) 1  q dζ (1 + |ζ  | + |ζn |)



1/2

228

6 Lp Theory of Elliptic Boundary Value Problems −q+n−1

= (1 + |ξn |)



Rn−1 −q+n−1

= C (1 + |ξn |)

1 dη  (1 + |η  |)q

for −q < −(n − 1).

Hence we have the inequality 

2

Y (·, ξn 0 ≤ C

Rn−1

|Z(ζ  , ζn )| dζ 

2

2

v s

(6.29)

−q+n−1 2 ≤ C v s 1 + |ξn |2 . Therefore, by inequalities (6.28) and (6.29) it follows that 



W 0 ≤ C v s ≤ C  v s

R

 s−μ−1/2−q+n−1 1 + |ξn |2 dξn

1/2 (6.30)

for 2(s − μ − 1/2 − q + n − 1) < −1.

By combining (6.18), (6.19) and (6.30), we obtain that |U, ϕ| ≤ C ϕ −s+μ+1/2 V 0 ≤ C ϕ −s+μ+1/2 W 0 ≤ C ϕ −s+μ+1/2 v s

for all ϕ ∈ C0∞ (Rn ).

This proves the desired inequality (6.24). Now the proof of Theorem 6.2 is complete.   Remark 6.9. In view of Theorem 4.49, it follows that condition (6.12) is invariant under change of coordinates. Furthermore, it is easy to see that every parametrix for an elliptic differential operator satisfies condition (6.12). (II) Secondly, the next theorem covers volume potentials (see [17, Th´eor`eme (2.2.2)], [18, Th´eor`eme 2.9], [26, Chapitre V, Th´eor`eme 2.5] for p := 2): Theorem 6.10. Let A ∈ Lm cl (M ) be as in Theorem 6.2. Then we have the following two assertions (i) and (ii): (i) The operator

 AΩ : f −→ A(f 0 )Ω

is continuous on C ∞ (Ω) into itself. (ii) If 1 < p < ∞, then the operator AΩ extends to a continuous linear operator AΩ : H s,p (Ω) −→ H s−m,p (Ω) for s > −1 + 1/p and 1 < p < ∞ (see Figure 6.5 below).

6.1 Classical Potentials and Pseudo-Differential Operators

229

A

Ω H s,p (Ω) − −−− −→ H s−m,p (Ω)

−−−−→ C ∞ (Ω) − AΩ

C ∞ (Ω)

Fig. 6.5. The mapping properties of the volume potential AΩ

Remark 6.11. The operator AΩ can be visualized as follows: AΩ : C ∞ (Ω) −→ D (M ) −→ D (M ) −→ C ∞ (Ω), A

where the first arrow is the zero extension to M and the last one is the restriction to Ω. Part (i) of Theorem 6.10 asserts that the volume potential AΩ preserves smoothness up to the boundary ∂Ω. Proof. We only prove part (i), by using the Fourier transform ([26, Chapitre V, Th´eor`eme 2.5]). (i) Let E : C ∞ (Ω) −→ C ∞ (Rn ) be Seeley’s extension operator (see Theorem 4.11). If f ∈ C ∞ (Ω), we let  Ef (x) for x ∈ Ω  , u(x) = 0 for x ∈ Ω, where Ω  is the exterior domain of Ω Ω  = Rn \ Ω. Then we have the formulas and

f 0 = Ef − u,

  A f 0 Ω = A(Ef )|Ω − Au|Ω ,

with A(Ef ) ∈ C0∞ (Rn ). Hence it suffices to show that the mapping C ∞ (Ω  )  g −→ (Au)|Ω ∈ C ∞ (Ω) is continuous. Just as in the proof of Theorem 4.19 (formula (4.8)), we may assume that    1 i x ·ξ  ixn ξn   Bu(x) = e e a (x, ξ , ξn ) u  (ξ , ξn ) dξn dξ  , (2π)n−1 Rn−1 Γξ

6 Lp Theory of Elliptic Boundary Value Problems

230

where





u (ξ , ξn ) =

Rn

 =

f0(x , xn ) dx  −i x ·ξ  e

Rn−1

0

e

−i xn ξn

−∞

  0 f (x , xn ) dxn dx .

However, we remark that the function  0 e−i xn ξn f0(x , xn ) dxn −∞

admits an analytic extension into the upper complex half-plane Im ξn > 0 and (ξ  , ξn ) is continuous in Im ξn ≥ 0. Hence we find that the Fourier transform u  n−1 rapidly decreasing with respect to the variable ξ ∈ R , for Im ξn ≥ 0. In this way, we obtain that Bu|Rn ∈ C ∞ (Rn+ ). +

The proof of Theorem 6.10 is complete.  

6.2 Dirichlet Problem In this section we shall consider the Dirichlet problem in the framework of Sobolev spaces of Lp type. This is a generalization of the classical potential approach to the Dirichlet problem in terms of pseudo-differential operators. Let Ω be a bounded domain of Euclidean space Rn with smooth boundary ∂Ω. Its closure Ω = Ω ∪ ∂Ω is an n-dimensional, compact smooth manifold with boundary. We may assume that Ω is the closure of a relatively compact open subset Ω of an n-dimensional, compact smooth manifold M without boundary in which Ω has a smooth boundary ∂Ω ([1], [83]). This manifold  is the double of Ω (see Figure 4.1). M =Ω We let n n   ∂2u ∂u Au = aij (x) + bi (x) + c(x)u (6.31) ∂x ∂x ∂x i j i i,j=1 i=1 be a second-order, uniformly elliptic differential operator with real coefficients  of Ω such that: on the double M = Ω ∞ (1) The aij (x) 2 are the components of a C symmetric contravariant tensor of type 0 on M and there exists a constant a0 > 0 such that n 

aij (x)ξi ξj ≥ a0 |ξ|2

on T ∗ (M ),

i,j=1

where T ∗ (M ) is the cotangent bundle of M .

6.2 Dirichlet Problem

231

(2) bi ∈ C ∞ (M ) for 1 ≤ i ≤ n. (3) c ∈ C ∞ (M ) and c(x) ≤ 0 in Ω. Following Seeley [105] and [106], we let   N0 (A) := u ∈ C ∞ (M ) : supp u ⊂ Ω, Au = 0 in Ω . It is known (see [105, Theorem 7]) that N0 (A) is finite dimensional. It should be emphasized that all norms on the finite-dimensional space N0 (A) are equivalent. We remark (see [3, pp. 274–277] and [82, Theorem 3.20]) that all the sufficiently large eigenvalues of the Dirichlet problem for the differential operator A and its formal adjoint A∗ lie in the parabolic type region, as will be discussed in Section 6.7 (see assertion (6.74)). Hence, by considering A − λ1 and A∗ − λ1 for certain positive number λ1 we may assume that N0 (A) = N0 (A∗ ) = {0}.

(SC)

The next theorem states the existence of a volume potential for A, which plays the same role for A as the Newtonian potential plays for the Laplacian (cf. [105, Theorem 1], [106, p. 239, Theorem 2] and [114, Theorem 8.2.1]; [122, Theorem 7.29]): Theorem 6.12 (Seeley). Assume that condition (SC) is satisfied. Then we have the following two assertions (i) and (ii): (i) The operator A : C ∞ (M ) → C ∞ (M ) is bijective, and its inverse C is a classical, elliptic pseudo-differential operator of order −2 on M . (ii) The operators A and C extend respectively to isomorphisms A

H s,p (M ) −−−−→ H s−2,p (M )       H s,p (M ) ←−−−− H s−2,p (M ) C

for all s ∈ R, which are still inverses of each other. Proof. First, by condition (SC) we can apply [106, Theorem 2] to find a classical, pseudo-differential operator C1 of order −2 on M such that (AC1 ) f = A (C1 f ) = f

for all f ∈ C ∞ (M ).

Similarly, by applying [106, Theorem 2] to the adjoint A∗ we can find a classical, pseudo-differential operator C2 of order −2 on M such that (A∗ C2 ) g = A∗ (C2 g) = g

for all g ∈ C ∞ (M ).

Hence, by passing to the adjoint we have the formula

6 Lp Theory of Elliptic Boundary Value Problems

232

∗

C2 ∗ f = C2 ∗ (AC1 f ) = (C2 ∗ A) (C1 f ) = (A∗ C2 ) (C1 f ) = C1 f for all g ∈ C ∞ (M ), and so ∗

(C1 A) g = C1 (Ag) = C2 ∗ (Ag) = ((A∗ C2 )) g = g

for all g ∈ C ∞ (M ).

Summing up, we have proved that C = C1 = C2 ∗ is the inverse of A. The proof of Theorem 6.12 is complete.   Next we construct a surface potential for A, which is a generalization of the classical Poisson kernel for the Laplacian. We let Kv = γ0 (C (v ⊗ δ∂Ω )) = C (v ⊗ δ∂Ω )|∂Ω

for v ∈ C ∞ (∂Ω).

Here v ⊗ δ∂Ω = v(x ) ⊗ δ(t) is a distribution on M defined by the formula v ⊗ δ∂Ω , ϕ · μ = v, ϕ(·, 0) · ω

for all ϕ(x , t) ∈ C ∞ (M ),

where μ is a strictly positive density on M and ω is a strictly positive density on ∂Ω = {t = 0}, respectively. Then we have the following theorem (cf. [114, Theorem 8.2.2]): Theorem 6.13. Assume that condition (SC) is satisfied. Then we have the following two assertions (i) and (ii): (i) The operator K is a classical elliptic pseudo-differential operator of order −1 on ∂Ω. (ii) The operator K : C ∞ (∂Ω) → C ∞ (∂Ω) is bijective, and its inverse L is a classical elliptic pseudo-differential operator of first order on ∂Ω. Furthermore, the operators K and L extend respectively to isomorphisms K

B σ,p (∂Ω) −−−−→ B σ+1,p (∂Ω)       B σ,p (∂Ω) ←−−−− B σ+1,p (∂Ω) L

for all σ ∈ R and 1 < p < ∞, which are still inverses of each other. Now we let Pϕ := C ((Lϕ) ⊗ δ∂Ω ) |Ω

for ϕ ∈ C ∞ (∂Ω).

Then the operator P maps C ∞ (∂Ω) continuously into C ∞ (Ω), and extends to a continuous linear operator P : B s−1/p,p (∂Ω) −→ H s,p (Ω)

6.2 Dirichlet Problem

for all s ∈ R. Furthermore, we have, for all ϕ ∈ B s−1/p,p (∂Ω),  APϕ = AC ((Lϕ) ⊗ δ∂Ω ) |Ω = ((Lϕ) ⊗ δ∂Ω ) |Ω = 0 γ0 (Pϕ) = Pϕ|∂Ω = γ0 (C ((Lϕ) ⊗ δ∂Ω )) = K (Lϕ) = ϕ

233

in Ω, on ∂Ω.

The operator P is called the Poisson operator or Poisson kernel. We let N (A, s, p) = {u ∈ H s,p (Ω) : Au = 0 in Ω} ,

s ∈ R.

Since the injection H s,p (Ω) → D (Ω) is continuous, it follows that the null space N (A, s, p) is a closed subspace of H s,p (Ω); hence it is a Banach space. Then we have the following fundamental result (cf. [105, Theorems 5 and 6]): Theorem 6.14 (Seeley). The Poisson operator P maps the Besov space B s−1/p,p (∂Ω) isomorphically onto the null space N (A, s, p) for all s ∈ R and 1 < p < ∞. More precisely, the spaces N (A, s, p) and B s−1/p,p (∂Ω) are isomorphic for all σ ∈ R and 1 < p < ∞ in such a way that P

B s−1/p,p (∂Ω) −−−−→ N (A, s, p)       B s−1/p,p (∂Ω) ←−−−− N (A, s, p) . γ0

By combining Theorems 6.12 and 6.14, we can obtain the following existence and uniqueness theorem for the Dirichlet problem (cf. [4], [50], [77], [133], [136], [146]): Theorem 6.15 (the Dirichlet case). Let 1 < p < ∞ and s > −2 + 1/p. If condition (SC) is satisfied, then the Dirichlet problem  Au = f in Ω, (6.32) γ0 u = u|∂Ω = ϕ on ∂Ω has a unique solution u(x) in H s,p (Ω) for any f ∈ H s−2,p (Ω) and any ϕ ∈ B s−1/p,p (∂Ω). Indeed, it suffices to note that the unique solution u of the Dirichlet problem (6.32) is given by the following formula: u = C(Ef )|Ω + P (ϕ − (CEf )|∂Ω )

in Ω.

(6.33)

Here E : H s,p (Ω) → H s,p (M ) is the Seeley extension operator (see Theorem 4.11).

234

6 Lp Theory of Elliptic Boundary Value Problems

Furthermore, in the Neumann problem for the differential operator A and its formal adjoint A∗ , we have parabolic condensation of eigenvalues along the negative real axis, as discussed in Agmon [3, pp. 276–277]. Hence, we replace A and A∗ by A − λ1 and A∗ − λ1 for certain positive number λ1 , respectively, we may assume that condition (SC) is satisfied. Therefore, we can prove the following existence and uniqueness theorem for the Neumann problem (cf. [4], [50], [77], [133], [136], [146]): Theorem 6.16 (the Neumann case). Let 1 < p < ∞ and s > −1 + 1/p. If condition (SC) is satisfied, then the Neumann problem ⎧ in Ω, ⎨Av =  g (6.4) ∂v  ⎩ =ψ on ∂Ω ∂n ∂Ω has a unique solution v(x) in H s,p (Ω) for any g ∈ H s−2,p (Ω) and any ψ ∈ B s−1−1/p,p (∂Ω). Here n is the unit inward normal to the boundary ∂Ω. By Theorem 6.16, we can introduce a linear operator GN : H s−2,p (Ω) −→ H s,p (Ω) as follows: For any g ∈ H s−2,p (Ω), the function v = GN g ∈ H s,p (Ω) is the unique solution of the homogeneous Neumann problem ⎧ in Ω, ⎨Av = g   (6.34) ∂v  ∂v  ⎩ γ1 v = =− = 0 on ∂Ω. ∂ν ∂Ω ∂n ∂Ω Here recall that ν = −n is the unit outward normal to the boundary ∂Ω (see Figure 1.2). The operator GN is called the Green operator for the Neumann problem.

6.3 Formulation of the Boundary Value Problem In this section we formulate elliptic boundary value problems in the framework of Lp Sobolev spaces. If u ∈ H 2,p (Ω) = W 2,p (Ω), we can define its traces γ0 u and γ1 u respectively by the formulas (see Theorems 4.17 and 4.18) ⎧ ⎨γ0 u = u|∂Ω ,  ∂u  ∂u  ⎩ γ1 u = =− . ∂ν ∂Ω ∂n ∂Ω Then we have the following theorem (cf. [2], [107]):

6.4 Special Reduction to the Boundary

235

Theorem 6.17 (the trace theorem). The trace map γ = (γ0 , γ1 ) : H 2,p (Ω) −→ B 2−1/p,p (∂Ω) ⊕ B 1−1/p,p (∂Ω) is continuous and surjective for all 1 < p < ∞. We define a first-order, boundary condition Bγu

   ∂u   := a(x ) + b(x )u = a(x )γ1 u + b(x )γ0 u ∂ν ∂Ω

(6.2) for u ∈ H 2,p (Ω).

Here: (1) a ∈ C ∞ (∂Ω) and a(x ) ≥ 0 on ∂Ω. (2) b ∈ C ∞ (∂Ω) and b(x ) ≥ 0 on ∂Ω. Then we have the following proposition: Proposition 6.18. The boundary operator Bγ : H 2,p (Ω) −→ B 1−1/p,p (∂Ω) is continuous for all 1 < p < ∞. Now we can formulate our boundary value problem for (A, Bγ) as follows: Given functions f ∈ Lp (Ω) and ϕ ∈ B 2−1/p,p (∂Ω), find a function u ∈ H 2,p (Ω) such that  Au = f in Ω, (6.1) Bγu = ϕ on ∂Ω.

6.4 Special Reduction to the Boundary In this section, by using the operators P and GN we shall show that the boundary value problem (6.1) can be reduced to the study of a pseudo-differential operator on the boundary. The virtue of this reduction is that there is no difficulty in taking adjoints or transposes after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints or transposes. This allows us to discuss the existence theory more easily. Here it should be emphasized that our reduction approach would break down if we use the Dirichlet problem (Theorem 6.15) as usual, instead of the Neumann problem (Theorem 6.16). First, we remark that every function u(x) in H 2,p (Ω) can be written in the following form: u(x) = v(x) + w(x), (6.35) where

236

6 Lp Theory of Elliptic Boundary Value Problems



v = GN (Au) ∈ H 2,p (Ω),   w = u − v ∈ N (A, 2, p) = z ∈ H 2,p (Ω) : Az = 0 in Ω .

Since the operator GN : Lp (Ω) → H 2,p (Ω) is continuous, it follows that the decomposition (6.35) is continuous; more precisely, we have the inequalities • v 2,p ≤ C Au p ≤ C u 2,p ; • w 2,p ≤ u 2,p + v 2,p ≤ C u 2,p . Here the letter C denotes a generic positive constant. Now we assume that u ∈ H 2,p (Ω) is a solution of the boundary value problem  Au = f in Ω, (6.1) Bγu = ϕ on ∂Ω. Then, by virtue of the decomposition (6.35) of u(x) this is equivalent to saying that w = u − v ∈ H 2,p (Ω) is a solution of the boundary value problem  Aw = 0 in Ω, (6.36) Bγw = ϕ − Bγv = ϕ − b(x ) γ0 (GN f ) on ∂Ω, since we have, by formulas (6.2) and (6.34), Bγv = a(x )γ1 (GN f ) + b(x )γ0 (GN f ) = b(x )γ0 (GN f )

on ∂Ω.

However, by Theorem 6.14 it follows that N (A, 2, p) and B 2−1/p,p (∂Ω) are isomorphic in such a way that: γ0

N (A, 2, p) −−−−→ B 2−1/p,p (∂Ω)       N (A, 2, p) ←−−−− B 2−1/p,p (∂Ω). P

Therefore, we find that w ∈ H 2,p (Ω) is a solution of problem (6.14) if and only if ψ(x ) ∈ B 2−1/p,p (∂Ω) is a solution of the equation Bγ (Pψ) = ϕ − b(x ) γ0 (GN f )

on ∂Ω.

(6.37)

Here ψ = γ0 w, or equivalently, w = Pψ. We remark that equation (6.37) is a generalization of the classical Fredholm integral equation. Summing up, we obtain the following proposition:

6.4 Special Reduction to the Boundary

237

Proposition 6.19. Let 1 < p < ∞. Assume that condition (SC) is satisfied. For given functions f ∈ Lp (Ω) and ϕ ∈ B 2−1/p,p (∂Ω), there exists a solution u ∈ H 2,p (Ω) of the boundary value problem (6.1) if and only if there exists a solution ψ(x ) ∈ B 2−1/p,p (∂Ω) of equation (6.37). Furthermore, the solutions u(x) and ψ(x ) are related as follows: u = GN f + Pψ, Bγ (Pψ) = ϕ − b(x ) γ0 (GN f )

on ∂Ω.

Remark 6.20. The advantage of equation (6.37) is that the terms in the righthand side have the same regularity ϕ − b(x ) γ0 (GN f ) ∈ B 2−1/p,p (∂Ω). We let T : C ∞ (∂Ω) −→ C ∞ (∂Ω) ϕ −→ Bγ (Pϕ) . Then we have, by formula (6.2), T = BγP = −a(x ) Π + b(x ), where Π is the Dirichlet-to-Neumann operator defined as follows (cf. [138, p. 134, formula (4.13)]:     ∂ ∂  Πϕ = (Pϕ) = − (Pϕ) ∂n ∂ν ∂Ω ∂Ω = −γ1 (Pϕ)

for all ϕ ∈ C ∞ (∂Ω).

It is known (cf. [26], [57], [61], [73], [93], [105], [133]) that the operator Π is a classical, elliptic pseudo-differential operator of first order on ∂Ω; hence the operator T = a(x )Π + b(x ) is a classical pseudo-differential operator of first order on the boundary ∂Ω. Consequently, Proposition 6.19 asserts that the boundary value problem (6.1) can be reduced to the study of the first-order pseudo-differential operator T on the boundary ∂Ω. We shall formulate this fact more precisely in terms of functional analysis. First, we remark that the operator T : C ∞ (∂Ω) → C ∞ (∂Ω) extends to a continuous linear operator T : B s,p (∂Ω) −→ B s−1,p (∂Ω),

s ∈ R.

Then we have the formula T ϕ = Bγ(Pϕ) for all ϕ ∈ B 2−1/p,p (∂Ω), since the Poisson operator

6 Lp Theory of Elliptic Boundary Value Problems

238

P : B 2−1/p,p (∂Ω) −→ N (A, 2, p) and the boundary operator Bγ : H 2,p (Ω) −→ B 1−1/p,p (∂Ω) are both continuous. We associate with the boundary value problem (6.1) a linear operator A = (A, Bγ) : H 2,p (Ω) −→ Lp (Ω) ⊕ B 2−1/p,p (∂Ω) as follows. (a) The domain D (A) of A is the space   D (A) = u ∈ H 2,p (Ω) = W 2,p (Ω) : Bγu ∈ B 2−1/p,p (∂Ω) . (b) Au = {Au, Bγu} for every u ∈ D (A). Note that the space B 2−1/p,p (∂Ω) is a right boundary space associated with the Dirichlet condition: a(x ) ≡ 0 and b(x ) ≡ 1 on ∂Ω. Since the operators A : H 2,p (Ω) −→ Lp (Ω) and

Bγ : H 2,p (Ω) → B 1−1/p,p (∂Ω)

are both continuous, it follows that A is a closed operator. Furthermore, the operator A is densely defined, since the domain D (A) contains the space C ∞ (Ω). The situation can be visualized in Figure 6.6 below. (A, B )

H 2,p (Ω) −−−−−→ Lp (Ω) ⊕ B 1−1/p,p (∂Ω)

D(A)

A

− −−−−→ Lp (Ω) ⊕ B 2−1/p,p (∂Ω)

C ∞ (Ω) −−−−−→ (A, B )

C ∞ (Ω) ⊕ C ∞ (∂Ω)

Fig. 6.6. The mapping property of the operator A

Similarly, we associate with equation (6.37) a linear operator T = BγP : B 2−1/p,p (∂Ω) −→ B 2−1/p,p (∂Ω) as follows.

6.4 Special Reduction to the Boundary

239

(α) The domain D (T ) is the space   D (T ) = ϕ ∈ B 2−1/p,p (∂Ω) : T ϕ ∈ B 2−1/p,p (∂Ω) . (β) T ϕ = T ϕ = Bγ (Pϕ) for every ϕ ∈ D (T ). Then the operator T is a densely defined, closed operator, since the operator T : B 2−1/p,p (∂Ω) −→ B 1−1/p,p (∂Ω) is continuous and since the domain D (T ) contains the space C ∞ (∂Ω). The situation can be visualized in Figure 6.7 below. T =B P

B 2−1/p,p (∂Ω) −−−−−→ B 1−1/p,p (∂Ω)

T

− −−−− → B 2−1/p,p (∂Ω)

D(T )

C ∞ (∂Ω)

−−−−−→ T =B P

C ∞ (∂Ω)

Fig. 6.7. The mapping property of the operator T

The next theorem states that A has regularity property if and only if T has. Theorem 6.21 (Regularity). Let 1 < p < ∞. The following two conditions (6.38) and (6.39) are equivalent: • u ∈ Lp (Ω), Au ∈ Lp (Ω) and Bγu ∈ B 2−1/p,p (∂Ω) =⇒ u ∈ H • ϕ∈B

2,p

(6.38)

(Ω).

−1/p,p

(∂Ω) and T ϕ ∈ B 2−1/p,p (∂Ω)

(6.39)

=⇒ ϕ ∈ B 2−1/p,p (∂Ω). Proof. (i) First, we show that assertion (6.38) implies assertion (6.39). To do this, assume that ϕ ∈ B −1/p,p (∂Ω)

and T ϕ ∈ B 2−1/p,p (∂Ω).

Then, by letting u = Pϕ we obtain that u ∈ Lp (Ω), Au = 0

and Bγu = T ϕ ∈ B 2−1/p,p (∂Ω).

6 Lp Theory of Elliptic Boundary Value Problems

240

Hence it follows from condition (6.38) that u ∈ H 2,p (Ω), so that, by Theorem 6.17, ϕ = γ0 u ∈ B 2−1/p,p (∂Ω). (ii) Conversely, we show that assertion (6.17) implies estimate (6.16). To do this, assume that u ∈ Lp (Ω), Au ∈ Lp (Ω)

and Bγu ∈ B 2−1/p,p (∂Ω).

Then the distribution u(x) can be decomposed as follows: u(x) = v(x) + w(x), where



v = GN (Au) ∈ H 2,p (Ω), w = u − v ∈ N (A, 0, p) = {z ∈ Lp (Ω) : Az = 0 in Ω} .

Moreover, Theorem 6.14 asserts that the distribution w(x) can be written in the form w = Pϕ, ϕ = γ0 w ∈ B −1/p,p (∂Ω). Hence we have, by Theorem 6.17, T ϕ = Bγ(Pϕ) = Bγw = Bγu − Bγv = Bγu − b(x )γ0 v ∈ B 2−1/p,p (∂Ω), since γ1 v = 0. Thus it follows from condition (6.39) that ϕ ∈ B 2−1/p,p (∂Ω), so that again, by Theorem 6.14, w = Pϕ ∈ H 2,p (Ω). This proves that

u = v + w ∈ H 2,p (Ω).

The proof of Theorem 6.21 is complete.   The next theorem states that a priori estimates for A are entirely equivalent to corresponding a priori estimates for T . Theorem 6.22 (Estimates). Let 1 < p < ∞. The following two estimates (6.40) and (6.41) are equivalent:  

u 2,p ≤ C Au p + |Bγu|2−1/p,p + u p for all u ∈ D (A). (6.40)   (6.41) |ϕ|2−1/p,p ≤ C |T ϕ|2−1/p,p + |ϕ|−1/p,p for all ϕ ∈ D (T ). Here and in the following the letter C denotes a generic positive constant.

6.4 Special Reduction to the Boundary

241

Proof. (i) First, we show that estimate (6.40) implies estimate (6.41). By taking u = Pϕ with ϕ ∈ D (T ) in estimate (6.40), we obtain that   (6.42)

Pϕ 2,p ≤ C |T ϕ|2−1/p,p + Pϕ p . However, Theorem 6.14 asserts that the Poisson operator P maps the Besov space B s−1/p,p (∂Ω) isomorphically onto the null space N (A, s, p) for all s ∈ R. Thus the desired estimate (6.41) follows from estimate (6.42). (ii) Conversely, we show that estimate (6.41) implies estimate (6.40). To do this, we express a function u ∈ D (A) in the form u(x) = v(x) + w(x), where



v = GN (Au) ∈ H 2,p (Ω),   w = u − v ∈ N (A, 2, p) = z ∈ H 2,p (Ω) : Az = 0 in Ω .

Then we have, by Theorem 6.16 with s := 2,

v 2,p = GN (Au) 2,p ≤ C Au p .

(6.43)

Furthermore, by applying estimate (6.41) to the distribution γ0 w we obtain that   |γ0 w|2−1/p,p ≤ C |T (γ0 w)|2−1/p,p + |γ0 w|−1/p,p   = C |Bγw|2−1/p,p + |γ0 w|−1/p,p   ≤ C |Bγu|2−1/p,p + |Bγv|2−1/p,p + |γ0 w|−1/p,p . In view of Theorem 6.14, this proves that  

w 2,p ≤ C |Bγu|2−1/p,p + |Bγv|2−1/p,p + w p   ≤ C |Bγu|2−1/p,p + |Bγv|2−1/p,p + u p + v p   ≤ C |Bγu|2−1/p,p + |Bγv|2−1/p,p + u p + v 2,p .

(6.44)

However, since γ1 v = 0, it follows from an application of Theorem 6.17 that |Bγv|2−1/p,p = |b(x )γ0 v|2−1/p,p ≤ C v 2,p .

(6.45)

Thus, by carrying estimates (6.43) and (6.45) into estimate (6.44) we obtain that   (6.46)

w 2,p ≤ C Au p + |Bγu|2−1/p,p + u p . Therefore, the desired estimate (6.40) follows from estimates (6.43) and (6.46), since u(x) = v(x) + w(x). The proof of Theorem 6.22 is complete.  

242

6 Lp Theory of Elliptic Boundary Value Problems

6.5 Boundary Value Problems via Boutet de Monvel Calculus In this section we consider a second-order, uniformly elliptic differential op of Ω such that erator A with real coefficients on the double M = Ω Au(x) =

n 

 ∂2u ∂u (x) + bi (x) (x) + c(x)u(x), ∂xi ∂xj ∂xi i=1 n

aij (x)

i,j=1

(6.31)

and a first-order, boundary condition B such that Bγu(x ) = a(x )

∂u  (x ) + b(x )u(x ) ∂ν

on ∂Ω.

(6.2)

Assume that the following three conditions (SC), (H.1) and (H.2) are satisfied: (SC) N0 (A) = N0 (A∗ ) = {0}. (H.1) a(x ) ≥ 0 and b(x ) ≥ 0 on ∂Ω. (H.2) a(x ) + b(x ) > 0 on ∂Ω. Now we can formulate the non-homogeneous Robin problem: Given functions f defined in Ω and ϕ defined on ∂Ω, respectively, find a function u in Ω such that  Au = f in Ω, (6.1) Bγu = ϕ on ∂Ω. In this section we study the general Robin problem (6.1) in the framework of Lp Sobolev spaces from the viewpoint of the Boutet de Monvel calculus, and prove index formulas of Agranoviˇc–Dynin type that relate the indices of two elliptic boundary value problems using the index of a pseudo-differential operator on the boundary (see [5], [67]). In this section, we derive an index formula of Agranoviˇc–Dynin type for the Neumann problem (6.4) and the general Robin problem (6.1) in the framework of Lp Sobolev spaces (Theorem 6.26). 6.5.1 Boundary Space Bs−1−1/p,p(∂Ω) First, we introduce a subspace of the Besov space B s−1−1/p,p (∂Ω) under the conditions (H.1) and (H.2), which is a suitable tool to investigate the general Robin boundary operator Bγ defined by formula (6.2). We let B (∂Ω) := a(x ) B s−1−1/p,p (∂Ω) + b(x ) B s−1/p,p (∂Ω) (6.47)   = ϕ = a(x )ϕ1 + b(x )ϕ2 : ϕ1 ∈ B s−1−1/p,p (∂Ω), ϕ2 ∈ B s−1/p,p (∂Ω) , s−1−1/p,p

and define a norm |ϕ|∗s−1−1/p,p

(6.48)

6.5 Boundary Value Problems via Boutet de Monvel Calculus

243

  = inf |ϕ1 |s−1−1/p,p + |ϕ2 |s−1/p,p : ϕ = a(x )ϕ1 + b(x )ϕ2 , where | · |t,p is the norm of the Besov space B t,p (∂Ω). Then we have the assertions B (∂Ω) = a(x ) B s−1−1/p,p (∂Ω) + b(x ) B s−1/p,p (∂Ω)  B s−1/p,p (∂Ω) if a(x ) ≡ 0 and b(x ) > 0 on ∂Ω, = s−1−1/p,p B (∂Ω) if a(x ) > 0 on ∂Ω. s−1−1/p,p

Indeed, it suffices to note that if a(x ) > 0 on ∂Ω, then any function ϕ ∈ B s−1−1/p,p (∂Ω) can be written in the form ϕ = a(x )

ϕ + b(x ) · 0, a(x )

ϕ ∈ B s−1−1/p,p (∂Ω). a(x )

Moreover, we have the following lemma (see [123, Lemma 6.8]): Lemma 6.23. Let 1 < p < ∞ and s > 1 + 1/p. Assume that the conditions (H.1) and (H.2) are satisfied. Then we have the two assertions: s−1−1/p,p

(i) The space B (∂Ω) is a Banach space with respect to the norm | · |∗s−1−1/p,p . (ii) For general a(x ), we have the continuous injections s−1−1/p,p

B s−1/p,p (∂Ω) ⊂ B

(∂Ω) ⊂ B s−1−1/p,p (∂Ω).

If u ∈ H s,p (Ω), we can define its traces γ0 u and γ1 u respectively by the formulas ⎧ ⎨γ0 u = u|∂Ω , ∂u  ⎩ γ1 u = , ∂ν ∂Ω and let γu = {γ0 u, γ1 u} . By applying Theorems 4.17 and 4.18 with j := 0 and j := 1, we obtain the following theorem: Theorem 6.24 (the trace theorem). Let 1 < p < ∞. Then the trace map γ = (γ0 , γ1 ) : H s,p (Ω) −→ B s−1/p,p (∂Ω) × B s−1−1/p,p (∂Ω) is continuous and surjective for every s > 1 + 1/p. Furthermore, we can prove the following trace theorem for the general Robin boundary operator Bγu = a(x ) γ1 u + b(x ) γ0 u.

(6.2)

244

6 Lp Theory of Elliptic Boundary Value Problems

Proposition 6.25. Let 1 < p < ∞. Then the general Robin boundary operator s−1−1/p,p (∂Ω) Bγ : H s,p (Ω) −→ B is continuous and surjective for every s > 1 + 1/p. Proof. We have only to prove the surjectivity of the operator Bγ. By virtue of Theorem 6.24, we obtain that, for any given function ϕ = a(x )ϕ1 + b(x )ϕ2 ∈ B

s−1−1/p,p

(∂Ω),

we can find a function u ∈ H s,p (Ω) such that  s−1/p,p γ0 u = ϕ2 ∈ B (∂Ω), s−1−1/p,p γ1 u = ϕ1 ∈ B (∂Ω). Then we have the formula Bγu = a(x )γ1 u + b(x )γ0 u = a(x )ϕ1 + b(x )ϕ2 s−1−1/p,p

= ϕ ∈ B

(∂Ω). s−1−1/p,p

This proves the surjectivity of Bγ : H s,p (Ω) → B The proof of Proposition 6.25 is complete.  

(∂Ω).

6.5.2 Index Formula of Agranoviˇ c–Dynin Type Index formulas of Agranoviˇc–Dynin type are rules that relate the indices of two elliptic boundary value problems using the index of a pseudo-differential operator on the boundary (see [5], [67]). In this subsection, we assume that condition (SC) is satisfied, and derive an index formula of Agranoviˇc–Dynin type for the Neumann problem and the Robin problem in the framework of Lp Sobolev spaces (Theorem 6.26). (I) First, we consider the non-homogeneous Robin problem  Au = f in Ω, (6.1)   Bγu = a(x ) γ1 u + b(x ) γ0 u = ϕ on ∂Ω. We express the Robin problem (6.1), in terms of the Boutet de Monvel calculus, as a mapping of the matrix form ⎛ ⎞ A 0 ⎠. B=⎝ Bγ 0 We remark that the mapping

6.5 Boundary Value Problems via Boutet de Monvel Calculus

⎛ B=⎝

A 0 Bγ 0

⎞ ⎠:

245

H s,p (Ω)

H s+2,p (Ω) −→ B s+1−1/p,p (∂Ω)

(6.49) s+1−1/p,p

B

is continuous for all s > −1 + 1/p with 1 < p σ(B) is homotopic to the following: ⎛   − ξ  2 + ν 2 ⎜ ⎝ −a(x ) iτ + b(x )

(∂Ω)

< ∞, and its principal symbol ⎞ 0

⎟ ⎠.

(6.50)

0

Here and in the following we use the notation ξ = (ξ  , ν) = (ξ1 , ξ2 , . . . , ξn−1 , ν) ∈ Rn , 6  ξ  = 1 + |ξ  |2 , ξ = (ξ  , ν) = (ξ1 , ξ2 , . . . , ξn−1 , ν) ∈ Rn 

ξ = (ξ , τ ) = (ξ1 , ξ2 , . . . , ξn−1 , τ ) ∈ R

n

for potential operators, for trace operators.

(II) Secondly, we consider the non-homogeneous Neumann problem ⎧ in Ω, ⎨Av = g  (6.51) ∂v  ⎩ γ1 v = = ψ on ∂Ω, ∂ν ∂Ω and express this Neumann problem, in terms of the Boutet de Monvel calculus, as a mapping of the matrix form ⎞ ⎛ A 0 ⎠. A=⎝ γ1 0 We remark that the mapping ⎞ ⎛ H s,p (Ω) H s+2,p (Ω) A 0 ⎠ ⎝ −→ : A= γ1 0 B s+1−1/p,p (∂Ω) B s+1−1/p,p (∂Ω)

(6.52)

is continuous for all s > −1 + 1/p with 1 < p < ∞, and its principal symbol σ(A) is homotopic to the following:  ⎞ ⎛  2 − ξ   + ν 2 0 ⎟ ⎜ (6.53) ⎠. ⎝ −iτ 0

246

6 Lp Theory of Elliptic Boundary Value Problems

(III) Thirdly, if we introduce the Fredholm boundary operator T by the formula T : C ∞ (∂Ω) −→ C ∞ (∂Ω) ϕ −→ Bγ (Pϕ) , then we have the formula T = −a(x )Π + b(x ).

(6.54)

Here it should be noticed that the Fredholm boundary operator T = −a(x )Π + b(x ) : B s+2−1/p,p (∂Ω) −→ B

s+1−1/p,p

(∂Ω)

is continuous for all s ∈ R and 1 < p < ∞. (IV) Now we can prove an index formula of Agranoviˇc–Dynin type for the Neumann problem (6.4) and the degenerate Robin problem (6.1) in the framework of Lp Sobolev spaces: Theorem 6.26. Assume that the conditions (SC), (H.1) and (H.2) are satisfied. The next index formula relates the indices of the operators B and A using the index of the Fredholm boundary operator T : ind B − ind A = ind T.

(6.55)

Proof. The proof is divided into four steps. Step 1: By applying Theorem 6.2 with a(x ) ≡ 1,

b(x ) ≡ 0,

we obtain that the homogeneous Neumann problem ⎧ in Ω, ⎨Av = f  ∂v  ⎩ γ1 v = = 0 on ∂Ω ∂ν 

(6.34)

∂Ω

has a unique solution v ∈ H [50], [136]). If we let

s+2,p

(Ω) for every function f ∈ H s,p (Ω) (see [4], v := GN f,

then it follows that the Green operator GN : H s,p (Ω) −→ H s+2,p (Ω) is continuous for all s > −1 + 1/p with 1 < p < ∞. Moreover, by applying Proposition 5.2 and formula (5.13) we find that the Dirichlet-to-Neumann operator Π, defined by the formula

6.5 Boundary Value Problems via Boutet de Monvel Calculus

  ∂ (Pϕ) Πϕ := ∂n

∂Ω

= −γ1 (Pϕ)

247

  ∂ (Pϕ) =− ∂ν ∂Ω

for all ϕ ∈ C ∞ (∂Ω),

maps B s+2−1/p,p (∂Ω) isomorphically onto B s+1−1/p,p (∂Ω) for all s ∈ R and 1 < p < ∞. Now we consider an operator D of the matrix form ⎞ ⎛ H s,p (Ω) H s+1,p (Ω) GN −P Π−1 ⎠ ⎝ : D= → (6.56) γ0 GN 0 B s+1−1/p,p (∂Ω) B s+1−1/p,p (∂Ω) and its principal symbol σ(D) is homotopic to the following (see Examples 5.2, 5.5 and 5.8): ⎛ 1 1 1 1 1 ⎞ 1 + 2 ξ   ξ   + iν ξ   − iτ ξ   + iν ξ   ⎟ ⎜ ξ  2 + ν 2 ⎜ ⎟ (6.57) ⎜ ⎟. ⎝ ⎠ 1 1 0 ξ   ξ   − iτ Step 2: By combining two formulas (6.52) and (6.56), we find that ⎛ ⎞⎛ ⎞ A 0 GN −P Π−1 ⎠⎝ ⎠ AD = ⎝ (6.58) γ0 GN 0 γ1 0 ⎛ ⎞ ⎛ ⎞ AGN −(AP) Π−1 I 0 ⎠=⎝ ⎠ =⎝ γ1 GN −(γ1 P) Π−1 0 Π Π−1 ⎛ ⎞ I 0 ⎠. =⎝ 0 I Therefore, the matrix form operator D is the right-inverse of the matrix form operator A, that is, ⎛ ⎞ H s,p (Ω) H s,p (Ω) I 0 ⎝ ⎠ AD = : −→ (6.59) 0 I B s+1−1/p,p (∂Ω) B s+1−1/p,p (∂Ω) Therefore, we obtain from formulas (6.58) and (6.59) that ⎛ ⎞ I 0 ⎠ = 0. ind A + ind D = ind ⎝ 0 I

(6.60)

Step 3: Furthermore, we obtain from formulas (6.49) and (6.56) that the composition of the matrix form operators B and D is equal to the following:

248

6 Lp Theory of Elliptic Boundary Value Problems

⎞ ⎛ ⎞ ⎞⎛ AGN −(AP) Π−1 A 0 GN −P Π−1 ⎠=⎝ ⎠ ⎠⎝ BD = ⎝ −1 γ0 GN 0 BγGN −(BγP) Π Bγ 0 ⎛ ⎞ I 0 ⎠ =⎝ b(x ) (γ0 GN ) (−a(x ) Π + b(x )) Π−1 ⎛ ⎞ I 0 ⎠, =⎝ b(x ) (γ0 GN ) −a(x ) + b(x ) Π−1 ⎛

(6.61)

since we have the formulas BγGN = a(x ) (γ1 GN ) + b(x ) (γ0 GN ) = b(x ) (γ0 GN ) , 







BγP = a(x ) (γ1 P) + b(x ) (γ0 P) = −a(x ) Π + b(x ).

(6.62a) (6.62b)

Moreover, we remark that the mapping H s,p (Ω)

H s,p (Ω) −→

BD : B s+1−1/p,p (∂Ω)

s+1−1/p,p

B

(∂Ω).

is continuous for all s > −1 + 1/p with 1 < p < ∞, Indeed, it suffices to note the following two assertions: • b(x )γ0 (GN f ) ∈ B s+2−1/p,p (∂Ω) ⊂ B

s+1−1/p,p

• −a(x )ψ + b(x )Π−1 ψ ∈ B

s+1−1/p,p

(∂Ω) for all f ∈ H s,p (Ω),

(∂Ω) for all ψ ∈ B s+1−1/p,p (∂Ω).

Step 4: Since the Dirichlet-to-Neumann operator Π : B s+2−1/p,p (∂Ω) −→ B s+1−1/p,p (∂Ω) is an isomorphism for all s ∈ R and 1 < p < ∞, we find from formulas (6.61) and(6.62) that ⎞ ⎛ I 0 ⎠ ind (B D) = ind ⎝ (6.63) b(x )γ0 GN −a(x ) + b(x ) Π−1   = ind −a(x ) + b(x ) Π−1 = ind (−a(x )Π + b(x )) = ind T. Therefore, by combining formulas (6.60) and (6.63) we obtain the desired index formula ind B − ind A = ind B + ind D = ind (B D) = ind T. The proof of Theorem 6.26 is complete.  

6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion

249

6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion In this section we study an intimate relationship between the Dirichlet-toNeumann operator and the reflecting diffusion in a bounded, domain D of Euclidean space RN with smooth boundary ∂D (Theorems 6.27 and 6.28 and Remark 6.29). We consider the following Dirichlet problem for the Laplacian Δ: For given function ϕ defined on ∂D, find a function u in D such that  Δu = 0 in D, (6.64) γ0 u = u|∂D = ϕ on ∂D. The existence and uniqueness theorem for the Dirichlet problem (6.42) is well established in the framework of H¨older spaces and Sobolev spaces (see [3], [4], [50], [77]). We denote its unique solution u as follows: u = H0 ϕ. The operator H0 is called the harmonic operator or Poisson operator for the Laplacian Δ. In this section we state that the pseudo-differential operator Π0 , defined by the formula   √ ∂ (H0 ϕ) = − −Λ ϕ for all ϕ ∈ C ∞ (∂D), Π0 ϕ := −γ1 (H0 ϕ) = ∂n ∂D is the generator of the Markov process on the boundary which is the trace of trajectories w of the reflecting diffusion (see Section 3.5)   X = xt , ζ, Bt , B, Px , x ∈ D ∪ {∂} . Here Λ is the Laplace–Beltrami operator on the boundary ∂D. The pseudodifferential operator Π0 is called the Dirichlet-to-Neumann operator (cf. formula (6.3)). More precisely, Sato and Ueno [98] constructed the following Markov process on the state space ∂D ∪ {∂} (see [98, Section 9]): Theorem 6.27 (Sato–Ueno). (1) Let H ∗ be the trace w∗ : [0, +∞] −→ ∂D ∪ {∂} on ∂D of trajectories w of the reflecting diffusion X . (2) A random variable ζ ∗ : H ∗ −→ [0, +∞], called the lifetime, defined by the formula

250

6 Lp Theory of Elliptic Boundary Value Problems

ζ ∗ (w∗ ) = τ −1 (ζ(w), w)

for w∗ = w|∂D ∈ H ∗ with w ∈ W .

Here ζ(w) is the lifetime of the reflecting diffusion X and τ −1 (t, w) is the right-continuous inverse of the local time τ (t, w) (see Section 3.6). We recall that Px ({w ∈ W : lim τ (t, w) = +∞}) = 1 t→+∞

(3) Let

if x ∈ D.

  x∗t (w∗ ) = xτ −1 (t,w) (w) = w τ −1 (t, w)

for w∗ = w|∂D ∈ H ∗ with w ∈ W (see Figure 6.8 below). We remark that   −1  ∈ ∂D for 0 ≤ t < ζ ∗ (w∗ ), ∗ ∗ xt (w ) = w τ (t, w) =∂ for ζ ∗ (w∗ ) ≤ t ≤ +∞, and that x∗t (w∗ ) is right-continuous with left limits for t ∈ [0, τ (+∞, w)].

∂D x∗τ (t,w) (w∗ ) = xt (w) τ (t, w) = 0

Fig. 6.8. The local time τ (t, w) and the trajectories x∗t (w∗ )

(4) Let Bt∗ = Bτ (t) and B ∗ = Bτ (∞) , respectively. (5) For each t ∈ [0, +∞], a pathwise shift mapping θt∗ : H ∗ −→ H ∗ defined by the formula θt∗ w∗ (s) = xτ (t+s,w) (w) = w (τ (t + s, w)) for all w∗ (t) = w|∂D ∈ H ∗ with coordinates xu (w) = w(u). (6) The system of measures {Px∗ , x ∈ ∂D ∪ {∂}} on (H ∗ , B ∗ ) such that   Px∗ (B) = Px {w ∈ W : xτ (t,w) (w) belongs to B as a function of t} for each B ∈ B ∗ = Bτ (∞) .

6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion

251

The Markov process X ∗ = (x∗t , H ∗ , Bt∗ , B ∗ , Px∗ , x ∈ ∂D ∪ {∂}) is called the Markov process on ∂D obtained from the reflecting diffusion   X = xt , W, Bt , B, Px, x ∈ D ∪ {∂} through time change by the local time τ (t) on the boundary (see Figure 6.9 below).

∂D Markov process X ∗ Reflecting diffusion X

Fig. 6.9. The Markov processes X and X ∗

The next theorem asserts that the Dirichlet-to-Neumann operator √ Π0 = − −Λ with principal symbol

− |ξ  |

is the generator of the Markov process X ∗ (see Volkonskiˇi [139], Sato–Ueno [98, Theorem 9.1], Ueno [137]): Theorem 6.28 (Sato–Ueno). (i) The Markov process X ∗ is conservative. Namely, we have the assertion Px∗ ({w∗ ∈ H ∗ : ζ ∗ (w∗ ) = +∞}) = 1

for each x ∈ ∂D.

(ii) A transition semigroup of linear operators     Tt∗ ϕ(x ) := Ex∗ (ϕ(x∗t )) = Ex ϕ xτ −1 (t) χ[0,τ (∞)) forms a Feller semigroup on the state space ∂D, and its infinitesimal generator is equal to the √ minimal closed extension Π0 of the Dirichlet-to-Neumann operator Π0 = − −Λ with the domain C 1 (∂D) in the Banach space C(∂D). More precisely, the infinitesimal generator Π0 is characterized as follows:

252

6 Lp Theory of Elliptic Boundary Value Problems

  (a) The domain of definition D Π0 is the space   D Π0 = {ϕ ∈ C(∂D) : Π0 ϕ ∈ C(∂D)} .   (b) Π0 ϕ = Π0 ϕ for every u ∈ D Π0 . We recall that Π0 ϕ is taken in the sense of distributions. Remark 6.29. By Example 4.8 with n := N − 1 and α := 1, we find that the principal √part of the distribution kernel of the Dirichlet-to-Neumann operator Π0 = − −Λ is equal to the following: Γ (N/2) 1  N/2 |x − y  |N π where |x − y  | is the geodesic distance between x and y  with respect to the Riemannian metric of ∂D induced by the natural metric of RN (see [46] and [122, Section 10.7]). Theorems 6.27 and 6.28 and Remark 6.29 can be visualized as in Table 6.1 below.

Infinitesimal generator

Distribution

Π0 = −γ1 H0

Γ (N/2) v.p. π N/2

1 |x − y  |N

kernel

Principal symbol

− |ξ  |

Dirichlet-to-Neumann operator

geodesic distance between x and y  minus the length of ξ 

Table 6.1. An overview of the reflecting diffusion process

For a given parameter α > 0, we study the following homogeneous Neumann boundary value problem: ⎧ in D, ⎨(Δ − α) u = f ∂u  ⎩ γ1 u = − = 0 on ∂D. ∂n ∂D To do so, we take a probability measure P (·) on the interval [0, +∞] with density α e−α t . Let P x be the product measure of Px and P on the product

6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion

253

space Ω = W × [0, +∞]. We define a sample path xt (ω) for ω = (w, T ) ∈ Ω and t ∈ [0, +∞] by the formula  xτ −1 (t,w) (w) if τ −1 (t, w) < T , xt (ω) = xt (w, T ) = ∂ if τ −1 (t, w) ≥ T , and also by its life time ζ(ω) by the formula ζ(ω) = inf {t ≥ 0 : xt (ω) = ∂} . Then we have the following assertion:  P x {xt (ω) is right-continuous with left limits as a function  of t ∈ [0, ζ(ω))} = 1 for each x ∈ D. Moreover, we define a measure Px on the space (H ∗ , B ∗ ) as follows: Px (B) = P x ({xt (ω) belongs to B as a function of t}) for each B ∈ B ∗ = Bτ (∞) .

The Markov process X  = (x∗t , H ∗ , Bt∗ , B ∗ , Px , x ∈ ∂D ∪ {∂}) is called the Markov process on ∂D obtained from the reflecting diffusion X = (xt , W, Bt , B, Px, x ∈ D ∪ {∂}) through time change by τ (t) and killing rate α. A transition semigroup of linear operators     −1 Tt ϕ(x ) := Ex  (ϕ(x∗t )) = Ex ϕ xτ −1 (t) e−α τ (t) χ[0,τ (∞)) forms a Feller semigroup on the state space ∂D, and its infinitesimal generator is equal to the minimal closed extension Πα of the Dirichlet-to-Neumann operator, defined by the formula   ∂ Πα ϕ = −γ1 (Πα ϕ) = (Hα ϕ) for all ϕ ∈ C ∞ (∂D), ∂n ∂D in the Banach space C(∂D), where Hα is the harmonic operator (or Poisson operator) for the operator Δ − α, that is, the function u = Hα ϕ is the unique solution of the Dirichlet problem  (Δ − α) u = 0 in D, γ0 u = u|∂D = ϕ on ∂D. More precisely, the infinitesimal generator Πα is characterized as follows:

254

6 Lp Theory of Elliptic Boundary Value Problems

  (a) The domain of definition D Πα is the space   D Πα = {ϕ ∈ C(∂D) : Πα ϕ ∈ C(∂D)} .   (b) Πα ϕ = Πα ϕ for every u ∈ D Πα . Here Πα ϕ is taken in the sense of distributions.

6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem Let Ω be a bounded domain in Euclidean space Rn with smooth boundary ∂Ω. Its closure Ω = Ω ∪ ∂Ω is an n-dimensional, compact smooth manifold with boundary. We let Au =

n  i,j=1

 ∂2u ∂u + bi (x) + c(x)u ∂xi ∂xj ∂x i i=1 n

aij (x)

be a second-order, elliptic differential operator with real coefficients such that: (1) aij ∈ C θ (Ω) with 0 < θ < 1, aij (x) = aji (x) for all x ∈ Ω and 1 ≤ i, j ≤ n, and there exists a constant a0 > 0 such that n 

aij (x)ξi ξj ≥ a0 |ξ|2

for all x ∈ Ω and ξ ∈ Rn .

(6.65)

i,j=1

(2) bi ∈ C(Ω) for 1 ≤ i ≤ n. (3) c ∈ C(Ω) and c(x) ≤ 0 in Ω. In this section we study the following Dirichlet eigenvalue problem: Given a complex parameter λ, find a function u(x) in Ω such that  −Au = λu in Ω, (∗) u=0 on ∂Ω, where λ is a complex spectral parameter. Following Mizohate [82, Chapter 3] and Wloka [146, Section 13], we introduce a closed linear operator A : L2 (Ω) −→ L2 (Ω) associated with the Dirichlet problem (∗) as follows: (1) The domain D(A) of definition is the space (see [80, Theorem 3.40])   D (A) = H01 (Ω) ∩ H 2 (Ω) = u ∈ H 2 (Ω) : γ0 u = 0 on ∂Ω .

6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem

255

(2) Au = Au for every u ∈ D(A). We remark that the Dirichlet eigenvalue problem (∗) is equivalent to the eigenvalue problem of the closed operator −A: − Au = λu.

(∗∗)

The purpose of this section is to prove that all the sufficiently large eigenvalues of the closed operator −A lie in the parabolic type region (see assertion (6.52) and Figure 6.10 below). 6.7.1 Unique Solvability of the Dirichlet Problem For all u, v ∈ H01 (Ω), we introduce a Dirichlet form A0 [u, v] by the formula A0 [u, v] =

n   i,j=1

aij (x)

Ω

∂u ∂v · dx. ∂xi ∂xj

(6.66)

We remark that if u, v ∈ C0∞ (Ω), then we have, by integration by parts, (Au + λu, v)L2 (Ω) (6.67)     n n  ∂aij ∂u + c(x)u + λu, v bi (x) − = −A0 [u, v] + ∂xj ∂xi L2 (Ω) i=1 j=1   n 0bi (x) ∂u + c(x)u + λu, v = −A0 [u, v] + , ∂xi L2 (Ω) i=1 where 0bi (x) := bi (x) −

n  ∂aij j=1

∂xj

for 1 ≤ i ≤ n.

Then we can prove the following existence and uniqueness theorem for the Dirichlet problem (∗): Theorem 6.30. Let λ ∈ C. Assume that there exists a constant C(λ) > 0 such that the inequality     2 (6.68) (Au + λu, u)L2 (Ω)  ≥ C(λ) u L2 (Ω) holds true for all u ∈ H01 (Ω) ∩ H 2 (Ω). Then the Dirichlet problem (∗) has a unique solution u ∈ H01 (Ω) ∩ H 2 (Ω) for any function f ∈ L2 (Ω). This implies that λ belongs to the resolvent set ρ (−A) of −A (see equation (∗∗)) λ ∈ ρ (−A) = −ρ (A) .

6 Lp Theory of Elliptic Boundary Value Problems

256

Proof. The proof is divided into three steps. Step (1): First, we decompose the differential operator A as follows: Au =

n  i,j=1

 ∂2u ∂u + bi (x) + c(x)u ∂xi ∂xj ∂x i i=1 n

aij (x)

:= A0 u + B0 u + c(x)u ⎛ ⎞    n n n   ∂ ∂a ∂u ij ⎠ ∂u ⎝bi (x) − + c(x)u, = aij (x) + ∂xj ∂xi ∂xj ∂xi i,j=1 i=1 j=1 where   ∂ ∂u A0 u = aij (x) , ∂xj ∂xi i,j=1 ⎛ ⎞ n n n    ∂aij ⎠ ∂u 0bi (x) ∂u . ⎝bi (x) − B0 u = = ∂x ∂x ∂xi j i i=1 j=1 i=1 n 

We remark that A0 is a formally self-adjoint, differential operator and further that B0 is a first-order differential operator. Hence, it follows from an application of the Rellich–Kondrachov theorem (Theorem 4.10) that the operator B0 + c(x) : H 2 (Ω) −→ H 1 (Ω) −→ L2 (Ω) is compact. However, we know from Gohberg–Kre˘ın [51, Theorem 2.6] and Schechter [100, Theorem 5.10] (see also Theorem 4.59) that the index is stable under compact perturbations. Therefore, we have the index formula ind A = ind A0 .

(6.69)

Step (2): On the other hand, there is a homotopy between the formally self-adjoint operator A0 and Laplacian Δ in the second-order differential operators. For example, we can take At = (1 − t) A0 + t Δ for 0 ≤ t ≤ 1. It is known (see [86, p. 187, Theorem 3], [146, Section 13, Theorem 13.2]) that the index of the Dirichlet problem is homotopy invariant in the class of second-order, formally self-adjoint differential operators: ind A0 = ind A1 .

(6.70)

Therefore, we are reduced to the study of the Dirichlet problem for the Laplacian:

6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem



−Δu = λu u=0

257

in Ω, on ∂Ω.

Step (3) However, we know from [146, Section 13, Theorem 13.5] (see also Corollary 4.63) that the index of the Dirichlet problem for the Laplacian is equal to zero in the space H01 (Ω) ∩ H 2 (Ω): ind A1 = 0.

(6.71)

Hence we have, by formulas (6.47), (6.48) and (6.49), ind A = ind A0 = ind A1 = 0.

(6.72)

In this way, we find that the index of the original Dirichlet eigenvalue problem (∗) is also equal to zero in the space H01 (Ω) ∩ H 2 (Ω). Therefore, if the fundamental inequality (6.68) holds true for some complex number λ, then we obtain from formula (6.72) that the Dirichlet eigenvalue problem (∗) has a unique solution u in the space H01 (Ω) ∩ H 2 (Ω) for any function f ∈ L2 (Ω). The proof of Theorem 6.30 is complete.   6.7.2 A Characterization of the Resolvent set of the Dirichlet Problem In this subsection we characterize the complex parameters λ for which the fundamental inequality (6.68) holds true. The next theorem asserts that the resolvent set ρ (−A) of −A contains the outside of a parabola (see Figure 6.10 below): Theorem 6.31. Let λ = μ + iν,

i=

√ −1.

We can find two constants p > 0 and q > 0 such that if the condition 2

(|ν| − p) ≥ 4p (μ + q)

(6.73)

is satisfied, then the desired inequality (6.46) holds true. Therefore, the Dirichlet eigenvalue problem (∗) has a unique solution u ∈ H01 (Ω) ∩ H 2 (Ω) for any function f ∈ L2 (Ω). In particular, we have the assertion   2 ρ (−A) = −ρ (A) ⊃ λ = μ + iν : (|ν| − p) ≥ 4p (μ + q) . (6.74) Proof. We consider the case where μ > 0,

258

6 Lp Theory of Elliptic Boundary Value Problems ν {λj } p −q

μ

O −p

  Fig. 6.10. The set ρ (−A) = −ρ (A) contains (|ν| − p)2 ≥ 4p (μ + q)

and prove that if the condition (6.73) holds true, then there exists a constant C(λ) > 0 such that the fundamental inequality (6.68) holds true. The proof is divided into three steps. Step (1): First, we have, by inequality (6.65), n   ∂u ∂u − (A0 u, u)L2 (Ω) = A0 [u, u] = aij (x) dx (6.75) ∂x i ∂xj i,j=1 Ω 2 n     ∂u    dx (x) ≥ a0  ∂xi  i=1 Ω  n    ∂u 2   . = a0  ∂xi  2 L (Ω) i=1 For any ε > 0, we have, by Schwarz’s inequality and inequality (6.75),       n  n      ∂u ∂u     i i 0b (x) ,u ,u (6.76) ≤  0b (x)       ∂x ∂x 2 2 i i L (Ω) L (Ω) i=1 i=1 n   ∂u     0i    max b (x) 

u L2 (Ω) ≤   2 ∂x x∈Ω i L (Ω) i=1  n    ∂u    ≤ C0

u L2 (Ω)  ∂xi  2 L (Ω) i=1 ? @ n    ∂u 2 √ @   ≤ C0 n A

u L2 (Ω)  ∂xi  2 L (Ω) i=1 C1 ≤ √ (A0 [u, u])1/2 u L2(Ω) a0 C2 2

u L2 (Ω) for all ε > 0, ≤ ε A0 [u, u] + ε

6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem

with

     C0 = max max 0bi (x) , 1≤i≤n x∈Ω √ C1 = n C0 ,

259

(6.77a) (6.77b)

2

C2 =

C1 n = C2. 4a0 4a0 0

(6.77c)

Here we have used the elementary inequality 2ab ≤ ε a2 +

1 2 b ε

for all ε > 0.

On the other hand, we have, by formula (6.67), Re (Au + λu, u)L2 (Ω) = −A0 [u, u] + μ u 2L2(Ω) + K1 ,

(6.78a)

Im (Au + λu, u)L2 (Ω) = ν u 2L2 (Ω) + K2 .

(6.78b)

Here it is easy to see the following two formulas:  n

 ∂u i 0 • K1 := Re + c(x)u, u b (x) ∂xi L2 (Ω) i=1      n i ∂0b 1 = − , + c(x) u, u 2 i=1 ∂xi L2 (Ω)   n 0bi (x) ∂u , u . • K2 := Im ∂xi L2 (Ω) i=1 Indeed, it suffices to note that we have, by integration by parts,     ∂u 0i 0bi (x) ∂u , u = , b (x) u ∂xi ∂xi L2 (Ω) L2 (Ω)     i 0 ∂b ∂u = − u, u − u, 0bi (x) ∂xi L2 (Ω) ∂xi L2 (Ω)  0i    ∂b ∂u i 0 =− u, u − b (x) ,u , ∂xi ∂xi L2 (Ω) L2 (Ω) since the 0bi (x) are real-valued. Hence we have the formula   ∂u i 0 ,u Re b (x) ∂xi L2 (Ω)      1 ∂u ∂u i i 0 0 = u, u + b (x) ,u b (x) 2 ∂xi ∂xi L2 (Ω) L2 (Ω)

6 Lp Theory of Elliptic Boundary Value Problems

260

  1 ∂0bi =− u, u . 2 ∂xi L2 (Ω) Moreover, by using inequality (6.76) and Schwarz’s inequality, we can estimate the terms K2 and K1 as follows:     n    ∂u   0bi (x) • |K2 | = Im ,u (6.79a)   ∂x i L2 (Ω)  i=1    n    ∂u   i 0 b (x) ,u ≤   ∂xi 2 L (Ω)  i=1

≤ ε A0 [u, u] +

C2

u 2L2(Ω) ε

for all ε > 0,

and • |K1 | ≤ C3 u 2L2 (Ω) ,   % $ n  1  ∂0bi   C3 = max − (x) + c(x) .  ∂xi x∈Ω  2

where

(6.79b)

(6.80)

i=1

Step (2): Now, by using formulas (6.78) we can rewrite the term (Au + λu, u)L2 (Ω) in the from (Au + λu, u)L2 (Ω) = Re (Au + λu, u)L2 (Ω) + i Im (Au + λu, u)L2 (Ω)     2 = −A0 [u, u] + μ u 2L2(Ω) + K1 + i ν u L2 (Ω) + K2 . However, we remark that if z = α + iβ, then we have the inequality ; |z| = α2 + β 2 ≥ (1 − t) |α| + t |β| for all 0 ≤ t ≤ 1.

(6.81)

Therefore, we obtain from inequalities (6.79) and (6.81) that     (6.82) (Au + λu, u)L2 (Ω)          ≥ (1 − t) −A0 [u, u] + μ u 2L2 (Ω) + K1  + t ν u 2L2 (Ω) + K2      ≥ (1 − t) A0 [u, u] − μ u 2L2(Ω) − |K1 | + t |ν| u 2L2 (Ω) − |K2 |   ≥ (1 − t) A0 [u, u] − μ u 2L2(Ω) − C3 u 2L2(Ω)

6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem

261

  C2

u 2L2(Ω) + t |ν| u 2L2(Ω) − ε A0 [u, u] − ε ≥ (1 − t − ε) A0 [u, u]     C2 + t |ν| − − (1 − t) (μ + C3 ) u 2L2 (Ω) . ε Step (3): In order to obtain the desired inequality (6.68), it suffices to prove the following two inequalities (6.83) and (6.84): 1−t−ε>0   C2 t |ν| − − (1 − t) (μ + C3 ) > 0 ε If we take ε=

(6.83) (6.84)

1−t , 2

then it follows that

  2C2 g(t) := t |ν| − − (1 − t) (μ + C3 ) 1−t 2C2 t = t (μ + |ν| + C3 ) − − μ − C3 . 1−t

Since we have the formula g  (t) = μ + |ν| + C3 −

2C2 , (1 − t)2

the function g(t) takes its positive maximum at the point B 2C2 t0 = 1 − , μ + |ν| + C3 under the condition μ + |ν| + C3 > 0. Then we have the formula g(t0 ) = |ν| − 2

(6.85)

; 2C2 (μ + |ν| + C3 ) + 2C2 .

Therefore, there exists a constant C(λ) > 0 in inequality (6.82) if we choose the complex number λ = μ + iν so large that ; g(t0 ) = |ν| − 2 2C2 (μ + |ν| + C3 ) + 2C2 > 0. Summing up, we have proved that if the inequality 2

(|ν| − p) ≥ 4p (μ + q)

(6.73)

holds true for the constants p = 2 C2 and q = C3 , then the two inequalities (6.83) and (6.84) hold true. The proof of Theorem 6.31 is complete.  

262

6 Lp Theory of Elliptic Boundary Value Problems

Remark 6.32. If the operator A is of the divergence form, that is, if the conditions n  ∂aij 0bi (x) = bi (x) − ≡ 0 in Ω for 1 ≤ i ≤ n ∂xj j=1 hold true, then we find from formulas (6.77), (6.80) and condition (6.85) that assertion (6.74) is reduced to the following: ρ (−A) = −ρ (A) ⊃ {λ = μ + iν : μ > −C3 , ν = 0} , where C3 = max |c(x)|. x∈Ω

6.8 Notes and Comments Agmon–Douglis–Nirenberg [4] and Lions–Magenes [77] are the classics for the general theory for higher order elliptic boundary value problems. See also Agmon [3], Agranovich–Vishik [6] and Peetre [89]. For more thorough treatments of this subject, the reader might refer to Chazarain–Piriou [26], H¨ormander [57], [61], Kumano-go [73], Seeley [105], [106], Taylor [133] and also Taira [122, Chapter 7]. Section 6.1: This section is an accessible and careful introduction to a modern version of the classical potential theory in terms of pseudo-differential operators. The material is taken from Boutet de Monvel [17], [18] and Chazarain– Piriou [26, Chapitre V, Section 2]. Theorems 6.2 and 6.10 for p = 2 correspond to Boutet de Monvel [17, Th´eor`emes (1.3.5) and (2.2.2)], respectively. See also Boutet de Monvel [18, Th´eor`emes 1.16 and 2.9]. Section 6.5: This section is devoted to the study of elliptic boundary value problems via the Boutet de Monvel calculus. The material is taken from Taira [126]. Moreover, this section is based on the talk entitled Probabilistic approach to pseudo-differential operators delivered at Special Seminar, Leibniz Universit¨ at Hannover, Germany, on October 16, 2019. Section 6.6: This section is inspired by a probabilistic approach due to Sato–Ueno [98], Ueno [137] and Watanabe [141]. This section is also based on the above talk at Leibniz Universit¨at Hannover. Section 6.7: The material is taken from Agmon [3, Sections 14 and 15], Mizohate [82, Chapter 3] and Wloka [146, Section 13]. The proof of Theorem 6.31 may be new. This chapter is an expanded and revised version of Chapter 4 of the second edition [121].

Part III

Analytic Semigroups in Lp Sobolev Spaces

7 Proof of Theorem 1.2

This chapter is devoted to the proof of Theorem 1.2. The idea of our proof is stated as follows. First, we reduce the study of the boundary value problem (∗)λ to that of a first-order pseudo-differential operator T (λ) = LP(λ) on the boundary ∂D, just as in Section 6.4. Then we prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate   (1.7)

u 2,p ≤ C(λ) f p + |ϕ|2−1/p,p + u p . More precisely, we construct a parametrix S(λ) for T (λ) in the H¨ ormander class L01,1/2 (∂D) (Lemma 7.2), and apply the Besov-space boundedness theorem (Theorem 4.47) to S(λ) to obtain the desired estimate (1.7) (Lemma 7.1). The proof of Theorem 1.2 can be flowcharted as in Table 7.1 below. Theorem 6.14 (Poisson operator) Proposition 6.19 (reduction to the boundary) Theorem 6.16 (Neumann problem) Theorem 1.2 (a priori estimate (1.7)) Lemma 7.2 (parametrix for T (λ)) Lemma 7.1 (a priori estimate (7.5)) Theorem 4.47 (Besov-space boundedness theorem)

Table 7.1. A flowchart for the proof of Theorem 1.2

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 7

266

7 Proof of Theorem 1.2

7.1 Boundary Value Problem with Spectral Parameter Let D be a bounded domain of Euclidean space RN with smooth boundary ∂D. Its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary. We may assume that D is the closure of a relatively compact  without open subset D of an N -dimensional, compact smooth manifold D  is the boundary in which D has a smooth boundary ∂D. This manifold D double of D (see Figure 7.1 below).

D

D

∂D

 of D Fig. 7.1. The double D

We let A=

N 

 ∂2 ∂ + bi (x) + c(x) ∂xi ∂xj ∂xi i=1 N

aij (x)

i,j=1

be a second-order, elliptic differential operator with real coefficients such that:  and aij (x) = aji (x) for all x ∈ D,  1 ≤ i, j ≤ N , and there (1) aij ∈ C ∞ (D) exists a positive constant a0 such that N 

aij (x)ξi ξj ≥ a0 |ξ|2

 on T ∗ (D),

i,j=1

 is the cotangent bundle of the double D.  where T ∗ (D) i ∞  (2) b ∈ C (D) for 1 ≤ i ≤ N .  and c(x) ≤ 0 in D. (3) c ∈ C ∞ (D) In this chapter we consider the elliptic boundary value problem (∗)λ with a spectral parameter  (A − λ) u = f in D, (∗)λ  ∂u  Lu = μ(x ) ∂n + γ(x )u = ϕ on ∂D. Here we recall that:

7.2 Proof of the A Priori Estimate (1.7)

267

(4) λ is a complex parameter. (5) μ(x ) and γ(x ) are real-valued, smooth functions on the boundary ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D (see Figure 1.1). The purpose of this chapter is to prove Theorem 1.2. More precisely, we prove the a priori estimate (1.7) provided that the following two conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D.

7.2 Proof of the A Priori Estimate (1.7) The proof of the a priori estimate (1.7) is divided into three steps. Step I: It suffices to show that estimate (1.7) holds true for some λ0 > 0, since we have, for all λ ∈ C, (A − λ0 ) u = (A − λ) u + (λ − λ0 ) u. We take a positive constant λ0 so large that the function c(x) − λ0 satisfies the condition  of D. (7.1) c(x) − λ0 < 0 on the double D This condition (7.1) implies that condition (SC) is satisfied for the operator A − λ0 : N0 (A − λ1 ) = N0 (A∗ − λ1 ) = {0}. (SC) Therefore, by applying Theorems 6.15 and 6.14 to the operator A − λ0 we can obtain the following two fundamental results (a) and (b): (a) The Dirichlet problem 

(A − λ0 ) w = 0 in D, w=ϕ on ∂D

(D)λ0

has a unique solution w(x) ∈ H t,p (D) for any function ϕ ∈ B t−1/p,p (∂D) with t ∈ R. (b) The Poisson operator P(λ0 ) : B t−1/p,p (∂D) −→ H t,p (D), defined by w = P(λ0 )ϕ, is an isomorphism of the space B t−1/p,p (∂D) onto the null space   N (A − λ0 , t, p) = u ∈ H t,p (D) : (A − λ0 )u = 0 in D for all t ∈ R; and its inverse is the trace operator γ0 on the boundary ∂D.

268

7 Proof of Theorem 1.2

We let T (λ0 ) : C ∞ (∂D) −→ C ∞ (∂D) ϕ −→ LP(λ0 )ϕ. Then we have the formula T (λ0 ) := LP(λ0 ) = μ(x ) Π(λ0 ) + γ(x ), where Π(λ0 ) is the Dirichlet-to-Neumann operator defined by the formula   ∂ (P(λ0 )ϕ) . Π(λ0 )ϕ = ∂n ∂D It is known that the operator Π(λ0 ) is a classical pseudo-differential operator of first order on the boundary ∂D and that its complete symbol is given by the following formula (cf. [122, Section 10.7]): √ √     p1 (x , ξ  ) + −1 q1 (x , ξ  ) + p0 (x , ξ  ) + −1 q0 (x , ξ  ) + terms of order ≤ −1 depending on λ0 , where p1 (x , ξ  ) < 0

(7.2) ∗

on the bundle T (∂D) \ {0} of non-zero cotangent vectors. For example, if A is the usual Laplacian Δ=

∂2 ∂2 ∂2 + + . . . + , ∂x21 ∂x22 ∂x2N

then we have the formula p1 (x , ξ  ) = minus the length |ξ  | of ξ  with respect to the Riemannian metric of ∂D induced by the natural metric of RN . Therefore, we obtain that the operator T (λ0 ) = μ(x ) Π(λ0 ) + γ(x ) is a classical pseudo-differential operator of first order on the boundary ∂D and further that its complete symbol t(x , ξ  ; λ0 ) is given by the following formula: √   (7.3) t(x , ξ  ; λ0 ) = μ(x ) p1 (x , ξ  ) + −1 q1 (x , ξ  ) √          + [γ(x ) + μ(x )p0 (x , ξ )] + −1 μ(x )q0 (x , ξ ) + terms of order ≤ −1 depending on λ0 .

7.2 Proof of the A Priori Estimate (1.7)

269

Then, by arguing just as in Section 6.4 we can prove that the question of the validity of a priori estimates and the question of regularity for solutions of problem (∗)λ for λ = λ0 are reduced to the corresponding questions for the operator T (λ0 ) (see Theorems 6.21 and 6.22). Step II: Therefore, in order to prove estimate (1.7) for λ = λ0 it suffices to show the following lemma: Lemma 7.1. Assume that conditions (A) and (B) are satisfied. Then we have, for all s ∈ R, ϕ ∈ D (∂D), T (λ0 )ϕ ∈ B s,p (∂D) =⇒ ϕ ∈ B s,p (∂D).

(7.4)

Furthermore, for any t < s, there exists a positive constant Cs,t such that |ϕ|s,p ≤ Cs,t (|T (λ0 )ϕ|s,p + |ϕ|t,p ) .

(7.5)

Proof. (a) The proof of Lemma 7.1 is based on the following lemma (cf. [68, Theorem 3.1]): Lemma 7.2. Assume that conditions (A) and (B) are satisfied. Then, for each point x of ∂D, we can find a neighborhood U (x ) of x such that: For any compact K ⊂ U (x ) and any multi-indices α, β, there exist positive constants CK,α,β and CK such that we have, for all x ∈ K and all |ξ  | ≥ CK ,    α β  −|α|+(1/2)|β| , Dξ Dx t(x , ξ  ; λ0 ) ≤ CK,α,β |t(x , ξ  ; λ0 )| (1 + |ξ|) |t(x , ξ  ; λ0 )|

−1

(7.6a)

≤ CK .

(7.6b)

Granting Lemma 7.2 for the moment, we shall prove Lemma 7.1. (b) First, we cover ∂D by a finite number of local charts m

{(Uj , χj )}j=1 in each of which inequalities (7.6a) and (7.6b)) hold true. Since the operator T (λ0 ) satisfies conditions (4.22a) and (4.22b) of Theorem 4.49 with μ := 0, ρ := 1 and δ = 1/2, it follows from an application of the same theorem that there exists a parametrix S(λ0 ) in the H¨ormander class L01,1/2 (Uj ) for T (λ0 ): 

T (λ0 )S(λ0 ) ≡ I S(λ0 )T (λ0 ) ≡ I

mod L−∞ (Uj ), mod L−∞ (Uj ).

m

m

Let {ϕj }j=1 be a partition of unity subordinate to the covering {Uj }j=1 , and choose a function ψj (x ) ∈ C0∞ (Uj ) such that ψj (x ) = 1 on supp ϕj , so that ϕj (x )ψj (x ) = ϕj (x ).

270

7 Proof of Theorem 1.2

Now we may assume that ϕ ∈ B t,p (∂D) for some t < s and that T (λ0 )ϕ ∈ B (∂D). We remark that the operator T (λ0 ) can be written in the following form: m m   ϕj T (λ0 )ψj + ϕj T (λ0 )(1 − ψj ). T (λ0 ) = s,p

j=1

j=1

However, by applying Theorems 4.42 and 4.36 to our situation we obtain that the second terms ϕj T (λ0 ) (1 − ψj ) are in L−∞ (∂D). Indeed, it suffices to note that ϕj (x ) (1 − ψj (x )) = ϕj (x ) − ϕj (x ) = 0. Hence we are reduced to the study of the first terms ϕj T (λ0 )ψj . This implies that we have only to prove the following local version of assertions (7.4) and (7.5): ψj ϕ ∈ B t,p (Uj ), T (λ0 )ψj ϕ ∈ B s,p (Uj ) =⇒ ψj ϕ ∈ B s,p (Uj ).    |ψj ϕ|s,p ≤ Cs,t |T (λ0 )ψj ϕ|2s,p + |ψj ϕ|2t,p .

(7.7) (7.8)

However, by applying the Besov-space boundedness theorem (Theorem 4.47) to our situation we find that the parametrix σ,p σ,p (Uj ) −→ Bloc (Uj ) S(λ0 ) : Bloc

is continuous for all σ ∈ R. This proves the desired assertions (7.7) and (7.8), since we have the assertion (see Theorem 4.36) ψj ϕ ≡ S(λ0 ) (T (λ0 )ψj ϕ)

mod C −∞ (Uj ).

Lemma 7.1 is proved, apart from the proof of Lemma 7.2.   Step III: Proof of Lemma 7.2 The proof of Lemma 7.2 is divided into five steps. Step III-1: First, we verify condition (7.6b): By assertions (7.3) and (7.2), we can find positive constants c0 and c1 such that we have, for all sufficiently large |ξ  |, |t(x , ξ  ; λ0 )| ≥ μ(x ) |p1 (x , ξ  ) + p0 (x , ξ  )| − γ(x )  c0 μ(x )|ξ  | − 12 γ(x ) if 0 ≤ μ(x ) ≤ c1 , ≥ c0    if c1 ≤ μ(x ) ≤ 1, 2 μ(x )|ξ | − γ(x ) since γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Hence we have, for all sufficiently large |ξ  |, |t(x , ξ  ; λ0 )| ≥ C (μ(x )|ξ  | + 1) . Here in the following the letter C denotes a generic positive constant. Inequality (7.9) implies the desired condition 7.6b:

(7.9)

7.2 Proof of the A Priori Estimate (1.7)

|t(x , ξ  ; λ0 )| ≥ C.

271

(7.10)

Step III-2: Next we verify condition (7.6a) for |α| = 1 and |β| = 0: Since we have, for all sufficiently large |ξ  |,   α     D  t(x , ξ ; λ0 ) ≤ C μ(x ) + |ξ  |−1 , ξ it follows from inequality (7.9) that  α    Dξ t(x , ξ ; λ0 ) ≤ C (1 + |ξ  |)−1 (μ(x )|ξ  | + 1) −1

≤ C (1 + |ξ  |)

|t(x , ξ  ; λ0 )| .

This inequality proves the desired condition (7.6a) for |α| = 1 and |β| = 0. Step III-3: We verify condition (7.6a) for |β| = 1 and |α| = 0: To do this, we need the following elementary lemma on non-negative functions: Lemma 7.3. Let f (x) be a non-negative, C 2 function on R such that we have, for some positive constant c, sup |f  (x)| ≤ c.

(7.11)

x∈R

Then we have the inequality |f  (x)| ≤

; ; 2c f (x)

on R.

(7.12)

Proof. In view of Taylor’s formula, it follows that 0 ≤ f (y) = f (x) + f  (x)(y − x) +

f  (ξ) (y − x)2 , 2

where ξ is between x and y. Thus, by letting z = x−y we obtain from estimate (7.11) that f  (ξ) 2 z 2 c ≤ f (x) + f  (x)z + z 2 for all z ∈ R. 2

0 ≤ f (x) + f  (x)z +

This implies the desired inequality (7.12).   Step III-4: Since we have, for all sufficiently large |ξ  |,       β    β Dx t(x , ξ  ; λ0 ) ≤ C Dx μ(x ) · |ξ  | + μ(x )|ξ  | + 1 , it follows from an application of Lemma 7.3 and inequalities (7.9) and (7.10) that    D C;  β  μ(x ) |ξ  | + 1 + (μ(x )|ξ  | + 1) Dx t(x , ξ  ; λ0 ) ≤ C

272

7 Proof of Theorem 1.2

C D 1/2 ≤ C |ξ  |1/2 (μ(x )|ξ  | + 1) + (μ(x )|ξ  | + 1)   −1/2 +1 ≤ C |t(x , ξ  ; λ0 )| |ξ  |1/2 |t(x , ξ  ; λ0 )| 1/2

≤ C |t(x , ξ  ; λ0 )| (1 + |ξ  |)

.

This inequality proves the desired condition (7.6a) for |β| = 1 and |α| = 0. Step III-5: Similarly, we can verify condition (7.6a) for the general case: |α| + |β| = k for k ∈ N. Now the proof of Lemma 7.1 and hence that of Theorem 1.2 is complete.  

7.3 Notes and Comments This chapter is a revised and expanded version of Chapter 5 of the second edition [121].

8 A Priori Estimates

This Chapter 8 and the next Chapter 9 are devoted to the proof of Theorem 1.4. In this chapter we study the operator Ap , and prove a priori estimates for the operator Ap − λI (Theorem 8.3) which will play a fundamental role in the next chapter. In the proof we make good use of Agmon’s method (Proposition 8.4). This is a technique of treating a spectral parameter λ as a second-order, elliptic differential operator of an extra variable and relating the old problem to a new problem with the additional variable.

8.1 A Priori Estimates via Agmon’s Method Recall that the operator Ap is a unbounded linear operator from Lp (D) into itself defined as follows: (a) The domain of definition D(Ap ) of Ap is the space D(Ap ) (1.8)   ∂u + γ(x )u = 0 on ∂D . = u ∈ H 2,p (D) = W 2,p (D) : Lu = μ(x ) ∂n (b) Ap u = Au for every u ∈ D(Ap ). We remark that the operator Ap is densely defined, since the domain D(Ap ) contains the space C0∞ (D). First, we have the following lemma: Lemma 8.1. Assume that the following two conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D.

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 8

274

8 A Priori Estimates

Then we have the a priori estimate  

u 2,p ≤ C Au p + u p

for all u ∈ D(Ap ).

(8.1)

Proof. The a priori estimate (8.1) follows immediately from estimate (1.7) of Theorem 1.2 with ϕ := 0.   Corollary 8.2. The operator Ap is a closed operator. Proof. Let {uj } be an arbitrary sequence in the domain D(Ap ) such that:  in Lp (D), uj −→ u Auj −→ v in Lp (D). Then, by applying estimate (8.1) to the sequence {uj } we find that {uj } is a Cauchy sequence in the space H 2,p (D). This proves that u ∈ H 2,p (D), and that uj −→ u

in H 2,p (D).

Hence we have the formula Au = lim Auj = v j→∞

in Lp (D),

and also, by Proposition 6.25 with B := L and s := 2, Lu = lim Luj = 0 in B 1−1/p,p (∂D). j→∞

Summing up, we have proved that u ∈ D(Ap ) and Ap u = v. The proof of Corollary 8.2 is complete.   The next theorem is an essential step in the proof of Theorem 1.4: Theorem 8.3. Assume that conditions (A) and (B) are satisfied. Then, for every −π < θ < π there exists a positive constant R(θ) depending on θ such that if λ = r2 ei θ and |λ| = r2 ≥ R(θ), we have, for all u ∈ H 2,p (D) satisfying Lu = 0 on ∂D (i. e., u ∈ D(Ap )), |u|2,p + |λ|

1/2

· |u|1,p + |λ| · u p ≤ C(θ) (A − λ) u p ,

(8.2)

with a positive constant C(θ) depending on θ. Here | · |j,p , j = 1, 2, is the seminorm on the space H 2,p (D) defined by the formula |u|j,p

⎛  =⎝



D |α|=j

⎞1/p |Dα u(x)|p dx⎠

.

8.1 A Priori Estimates via Agmon’s Method

275

Proof. The proof of Theorem 8.3 is divided into two steps. Step I: We shall make use of a method essentially due to Agmon (see [3], [45], [77], [113]). We introduce an auxiliary variable y of the unit circle S = R/2π Z, and replace the complex parameter λ by the second-order differential operator −ei θ

∂2 ∂y 2

for −π < θ < π.

Namely, we replace the operator A − λ by the operator ∂2 0 Λ(θ) := A + ei θ 2 ∂y

for −π < θ < π,

on the product domain D × S (see Figure 8.1 below), and consider instead of the problem with spectral parameter  (A − λ) u = f in D, (∗)λ  ∂u  Lu = μ(x ) ∂n + γ(x )u = 0 on ∂D the following boundary value problem:    0 u = A + ei θ ∂ 22 u Λ(θ)0 0 = f0 in D × S, ∂y ∂ u L0 u = μ(x ) ∂n + γ(x ) u 0=0

on ∂D × S.

(8.3)

0 We remark that the operator Λ(θ) is elliptic for −π < θ < π. D×S

D

Fig. 8.1. The product domain D × S

Then we have the following result, analogous to Lemma 8.1 (see [127, Theorem 9.1]):

276

8 A Priori Estimates

Proposition 8.4. Assume that conditions (A) and (B) are satisfied. Then we u = 0 on have, for all u 0 ∈ H 2,p (D × S) satisfying the boundary condition L0 ∂D × S,     0 0 u + 0

0 u ≤ C(θ) u p , (8.4) Λ(θ)0 2,p

p

0 with a positive constant C(θ) depending on θ. Proof. We reduce the study of the boundary value problem (8.3) to that of a pseudo-differential operator on the boundary ∂D × S, just as in problem (∗)λ . We can prove (see [127, Theorem 10.1]) that Theorems 6.15 and 6.14 remain valid for the differential operator ∂2 0 Λ(θ) = A + ei θ 2 ∂y

for −π < θ < π.

(0 a) The Dirichlet problem 

0 w Λ(θ) 0 = 0 in D × S, w 0=ϕ 0 on ∂D × S

0 (D)

has a unique solution w(x, 0 y) ∈ H t,p (D × S) for any function ϕ(x 0  , y) ∈ t−1/p,p B (∂D × S) with t ∈ R. (0b) The Poisson operator 0 : B t−1/p,p (∂D × S) −→ H t,p (D × S), P(θ) 0 ϕ, defined by w 0 = P(θ) 0 is an isomorphism of the space B t−1/p,p (∂D × S) onto the null space   0 0 u = 0 in D × S N (Λ(θ), t, p) = u 0 ∈ H t,p (D × S) : Λ(θ)0 for all t ∈ R; and its inverse is the trace operator on the boundary ∂D ×S. We let T0(θ) : C ∞ (∂D × S) −→ C ∞ (∂D × S) 0 ϕ. ϕ 0 −→ LP(θ) 0 Then the operator T0(θ) can be decomposed as follows: 0 0 T0(θ) := LP(θ) = μ(x ) Π(θ) + γ(x ),

(8.5)

0 where Π(θ) is the Dirichlet-to-Neumann operator defined by the formula

8.1 A Priori Estimates via Agmon’s Method

 ∂ 0 0 ϕ P(θ)ϕ 0  Π(θ) 0= ∂n ∂D×S

277

for ϕ 0 ∈ C ∞ (∂D × S).

0 It is known that the operator Π(θ) is a classical pseudo-differential operator of first order on the boundary ∂D × S and further that its complete symbol is given by the following formula (cf. [122, Section 10.7]): √   p01 (x , ξ  , y, η; θ) + −1 q01 (x , ξ  , y, η; θ) √   + p00 (x , ξ  , y, η; θ) + −1 q00 (x , ξ  , y, η; θ) + terms of order ≤ −1, where p01 (x , ξ  , y, η; θ) < 0 on the bundle T ∗ (∂D × S) \ {0} of non-zero cotangent vectors, for −π < θ < π.

(8.6)

For example, if A is the usual Laplacian Δ=

∂2 ∂2 ∂2 + 2 + ... + 2 , 2 ∂x1 ∂x2 ∂xN

then we have the formula p01 (x , ξ  , y, η; θ) ⎡ C D1/2    ⎤1/2 2  2 2 2 4  2 2 | + cos θ · η + sin θ · η + |ξ | + cos θ · η |ξ ⎥ ⎢ = −⎣ ⎦ . 2 Therefore, we obtain that the operator 0 T0(θ) = μ(x ) Π(θ) + γ(x ) is a classical pseudo-differential operator of first order on the boundary ∂D×S and further that its complete symbol is given by the following formula: √   μ(x ) p01 (x , ξ  , y, η; θ) + −1 q01 (x , ξ  , y, η; θ) (8.7) √          p0 (x , ξ , y, η; θ)] + −1 μ(x )0 q0 (x , ξ , y, η; θ) + [γ(x ) + μ(x )0 + terms of order ≤ −1. Then, by virtue of assertions (8.7) and (8.6) we can verify that the operator T0(θ) satisfies conditions (4.22a) and (4.22b) of Theorem 4.49 with μ := 0, ρ := 1 and δ := 1/2, just as in the proof of Lemma 7.2. Hence we obtain the following lemma, analogous to Lemma 7.1 (see [127, Proposition 12.3]):

278

8 A Priori Estimates

Lemma 8.5. Assume that conditions (A) and (B) are satisfied. Then we have, for all s ∈ R, ϕ 0 ∈ D (∂D × S), T0(θ)ϕ 0 ∈ B s,p (∂D × S) =⇒ ϕ 0 ∈ B s,p (∂D × S). 0s,t such that Furthermore, for any t < s, there exists a positive constant C   0s,t |T0(θ)ϕ| 0 s,p + |ϕ| (8.8) 0 t,p . |ϕ| 0 s,p ≤ C The desired estimate (8.4) follows from estimate (8.8) with s := 2 − 1/p and t := −1/p, just as in the proof of Theorem 6.22. The proof of Proposition 8.4 is complete.   Step II: Now let u(x) be an arbitrary function in the domain D(Ap ): u ∈ H 2,p (D) and Lu = 0 on ∂D. We choose a function ζ(y) in C ∞ (S) such that ⎧ ⎪ 1 on S, ⎨0 ≤ ζ(y) ≤   supp ζ ⊂ π3 , 5π , 3 ⎪ ⎩ ζ(y) = 1 for π2 ≤ y ≤

3π 2 ,

and let v0η (x, y) = u(x) ⊗ ζ(y) ei η y

for x ∈ D, y ∈ S and η ≥ 0.

Then we have the assertions • 0 vη ∈ H 2,p (D × S),   2 0 vη = A + ei θ ∂ • Λ(θ)0 v0η ∂y 2 = (A − η 2 ei θ )u ⊗ ζ(y)ei η y + 2(iη)ei θ u ⊗ ζ  (y) ei η y + ei θ u ⊗ ζ  (y) ei η y , and also

L0 vη (x , y) = (Lu(x )) ⊗ ζ(y) ei η y = 0

on ∂D × S.

Thus, by applying inequality (8.4) to the functions v0η (x, y) = u(x) ⊗ ζ(y) ei η y

for x ∈ D, y ∈ S and η ≥ 0,

we obtain that

        0 iη y  0 u ⊗ ζ ei η y  ≤ C(θ) u ⊗ ζ ei η y  . Λ(θ) u ⊗ ζ e +   2,p p p

We can estimate each term of inequality (8.9) as follows:

(8.9)

8.1 A Priori Estimates via Agmon’s Method

 1/p   i η y p p  • u ⊗ ζe = |u(x)| |ζ(y)| dx dy = ζ p · u p . p D×S     0 u ⊗ ζ ei η y  • Λ(θ) p       ≤ (A − η 2 ei θ )u ⊗ ζei η y p + 2η u ⊗ ζ  ei η y p + u ⊗ ζ  ei η y p      ≤ ζ p ·  A − η 2 ei θ up + 2η ζ  p + ζ  p u p . p  • u ⊗ ζei η y 2,p    α   i η y p D = dx dy x,y u(x) ⊗ ζ(y) e |α|≤2

≥

3π/2

π/2

D

  

=

k+|β|≤2

≥π

(8.10) (8.11)

(8.12)

D×S

  

|α|≤2

279

D

  |β|=2

D

 α   Dx,y u(x) ⊗ ei η y p dx dy

3π/2

 k β  η D u(x)p dx dy

π/2

        β p D u(x)p dx + η p Dβ u(x)p dx + η 2p |u(x)| dx |β|=1

  p p p = π |u|2,p + η p |u|1,p + η 2p u p .

D

D

Therefore, by carrying these three inequalities (8.10), (8.11) and (8.12) into inequality (8.9) we obtain that    0  (θ) (A − η 2 ei θ )u + η u , |u|2,p + η |u|1,p + η 2 u p ≤ C p p 0 (θ) independent of η. with a positive constant C If η is so large that 0  (θ), η ≥ 2C then we can eliminate the last term on the right-hand side to obtain that   0  (θ) (A − η 2 ei θ )u . |u|2,p + η |u|1,p + η 2 u p ≤ 2 C p This proves the desired inequality (8.2)) if we take λ := η 2 ei θ , 0  (θ)2 , R(θ) := 4 C 0  (θ). C(θ) := 2 C The proof of Theorem 8.3 is now complete.  

280

8 A Priori Estimates

8.2 Notes and Comments This chapter is a revised and expanded version of Chapter 6 of the second edition [121]. For the detailed proof of Proposition 8.4, the reader is referred to Taira [127].

9 Proof of Theorem 1.4

In this chapter we prove Theorem 1.4 (Theorems 9.1 and 9.11). Once again we make use of Agmon’s method in the proof of Theorems 9.1 and 9.11. In particular, Agmon’s method plays an important role in the proof of the surjectivity of the operator Ap − λI (Proposition 9.2). The proof of Theorem 1.4 can be flowcharted as in Table 9.1 below. Theorem 8.3 (a priori estimate (8.2)) Theorem 9.1 (resolvent estimate (1.9)) Proposition 8.4 (a priori estimate (8.4)) Theorem 1.4 Theorem 9.11 Lemma 9.4 (dim N (T0p (θ)) < ∞) Proposition 9.2 (zero index) Lemma 9.9 (codim R(Tp (θ)) < ∞) Table 9.1. A flowchart for the proof of Theorem 1.4

9.1 Proof of Theorem 1.4, Part (i) First, we prove part (i) of Theorem 1.4: Theorem 9.1. Assume that the following two conditions (A) and (B) are satisfied: © Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 9

282

9 Proof of Theorem 1.4

(A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D. Then, for every 0 < ε < π/2 there exists a positive constant rp (ε) such that the resolvent set of Ap contains the set   Σp (ε) = λ = r2 ei θ : r ≥ rp (ε), −π + ε ≤ θ ≤ π − ε , and that the resolvent (Ap − λI)−1 satisfies the estimate   (Ap − λI)−1  ≤ cp (ε) |λ|

for all λ ∈ Σp (ε),

(1.9)

where cp (ε) is a positive constant depending on ε. Proof. The proof of Theorem 9.1 is divided into three steps. Step I: By estimate (8.2), it follows that if λ = r2 ei θ for −π < θ < π and if |λ| = r2 ≥ R(θ), then we have, for all u ∈ D(Ap ), |u|2,p + |λ|

1/2

· |u|1,p + |λ| · u p ≤ C(θ) (Ap − λI) u p .

However, we find from the proof of Theorem 8.3 that the constants R(θ) and C(θ) depend continuously on θ ∈ (−π, π), so that they may be chosen uniformly in θ ∈ [−π + ε, π + ε], for every ε > 0. This proves the existence of the constants rp (ε) and cp (ε), that is, we have, for all λ = r2 ei θ satisfying the conditions r ≥ rp (ε) and −π + ε ≤ θ ≤ π + ε, |u|2,p + |λ|

1/2

· |u|1,p + |λ| · u p ≤ cp (ε) (Ap − λI) u p .

(9.1)

By estimate (9.1), we find that the operator Ap − λI is injective and its range R(Ap − λI) is closed in the space Lp (D), for all λ ∈ Σp (ε). Step II: We show that the operator Ap − λI is surjective for all λ ∈ Σp (ε), that is, (9.2) R(Ap − λI) = Lp (D) for all λ ∈ Σp (ε). To do this, it suffices to show that the operator Ap −λI is a Fredholm operator with (9.3) ind (Ap − λI) = 0 for all λ ∈ Σp (ε), since Ap − λI is injective for all λ ∈ Σp (ε). Here we recall that a densely defined, closed linear operator T with domain D(T ) from a Banach space X into itself is called a Fredholm (closed) operator if it satisfies the following three conditions (i), (ii) and (iii): (i) The null space N (T ) = {x ∈ D(T ) : T x = 0} of T has finite dimension, that is, dim N (T ) < ∞. (ii) The range R(T ) = {T x : x ∈ D(T )} of T is closed in X. (iii) The range R(T ) has finite codimension in X, that is, codim R(T ) = dim X/R(T ) < ∞.

9.1 Proof of Theorem 1.4, Part (i)

283

In this case the index ind T of T is defined by the formula ind T = dim N (T ) − codim R(T ). Step II-1: We reduce the study of the operator Ap − λI for λ ∈ Σp (ε) to that of a pseudo-differential operator on the boundary, just as in the proof of Theorem 1.2. Let T (λ) be a classical pseudo-differential operator of first order on the boundary ∂D defined as follows: T (λ) = LP(λ) = μ(x ) Π(λ) + γ(x ) for λ ∈ Σp (ε),

(9.4)

where Π(λ) is the Dirichlet-to-Neumann operator defined by the formula Π(λ) : C ∞ (∂D) −→ C ∞ (∂D)

  ∂ ϕ −→ (P(λ)ϕ) . ∂n ∂D

Since the operator T (λ) : C ∞ (∂D) → C ∞ (∂D) extends to a continuous linear operator T (λ) : B t,p (∂D) −→ B t−1,p (∂D) for all t ∈ R, we can introduce a densely defined, closed linear operator Tp (λ) : B 2−1/p,p (∂D) −→ B 2−1/p,p (∂D) as follows. (α) The domain D (Tp (λ)) of Tp (λ) is the space   D (Tp (λ)) = ϕ ∈ B 2−1/p,p (∂D) : T (λ)ϕ ∈ B 2−1/p,p (∂D) . (β) Tp (λ)ϕ = T (λ)ϕ for every ϕ ∈ D (Tp (λ)). Then we can obtain the following three results (I), (II) and (III) (see [114, Section 8.3]): (I) The null space N (Ap − λI) of Ap − λI has finite dimension if and only if the null space N (Tp (λ)) of Tp (λ) has finite dimension, and we have the formula dim N (Ap − λI) = dim N (Tp (λ)) . (II) The range R(Ap − λI) of Ap − λI is closed if and only if the range R (Tp (λ)) of Tp (λ) is closed; and R(Ap − λI) has finite codimension if and only if R (Tp (λ)) has finite codimension, and we have the formula codim R (Ap − λI) = codim R (Tp (λ)) .

284

9 Proof of Theorem 1.4

(III) The operator Ap − λI is a Fredholm operator if and only if the operator Tp (λ) is a Fredholm operator, and we have the formula ind (Ap − λI) = ind Tp (λ). Therefore, the desired assertion (9.3) is reduced to the following assertion: ind Tp (λ) = 0 for all λ ∈ Σp (ε).

(9.5)

Step II-2: To prove assertion (9.5), we shall make use of Agmon’s method just as in Chapter 8. Let T0(θ) be the classical pseudo-differential operator of first order on the boundary ∂D × S introduced in Chapter 8 (see formula (8.5)): 0 0 + γ(x ) for −π < θ < θ, T0(θ) = LP(θ) = μ(x ) Π(θ) 0 where Π(θ) is the Dirichlet-to-Neumann operator defined by the formula 0 : C ∞ (∂D × S) −→ C ∞ (∂D × S) Π(θ)  ∂ 0 P(θ)ϕ˜  . ϕ˜ −→ ∂n ∂D×S

We define a densely defined, closed linear operator T0p (θ) : B 2−1/p,p (∂D × S) −→ B 2−1/p,p (∂D × S) as follows.

  (˜ α) The domain D T0p (θ) of T0p (θ) is the space     D T0p (θ) = ϕ˜ ∈ B 2−1/p,p (∂D × S) : T0(θ)ϕ˜ ∈ B 2−1/p,p (∂D × S) .   ˜ T0p (θ)ϕ˜ = T0(θ)ϕ˜ for every ϕ˜ ∈ D T0p (θ) . (β) Then the most fundamental relationship between the operators T0p (θ) and Tp (λ) is stated as follows: Proposition 9.2. If ind T0p (θ) is finite, then there exists a finite subset K of Z such that the operator Tp (λ ) is bijective for all λ = 2 ei θ satisfying  ∈ Z \ K. Granting Proposition 9.2 for the moment, we shall prove Theorem 9.1. Step III: End of Proof of Theorem 9.1 Step III-1: We show that if conditions (A) and (B) are satisfied, then we have the assertion ind T0p (θ) = dim N (T0p (θ)) − codim R(T0p (θ)) < ∞.

(9.6)

To this end, we need a useful criterion for Fredholm operators due to Peetre [89] (cf. [114, Theorem 3.7.6]):

9.1 Proof of Theorem 1.4, Part (i)

285

Lemma 9.3 (Peetre). Let X, Y , Z be Banach spaces such that X ⊂ Z is a compact injection, and let T be a closed linear operator with D(T ) from X into Y with domain D(T ). Then the following two conditions (i) and (ii) are equivalent: (i) The null space N (T ) of T has finite dimension and the range R(T ) of T is closed in Y . (ii) There is a positive constant C such that

x X ≤ C ( T x Y + x Z )

for all x ∈ D(T ).

(9.7)

Proof. (i) =⇒ (ii): Since the null space N (T ) has finite dimension, we can find a closed topological complement X0 in X: X = N (T ) ⊕ X0 .

(9.8)

This gives that D(T ) = N (T ) ⊕ (D(T ) ∩ X0 ) . Namely, every element x of D(T ) can be written in the form (see Figure 9.1 below) x = x0 + x1 , x0 ∈ D(T ) ∩ X0 , x1 ∈ N (T ). Moreover, since the range R(T ) is closed in Y , it follows from an application of the closed graph theorem (see [147, Chapter II, Section 6, Theorem 1]) that there exists a positive constant C such that

x0 X ≤ C T x0 Y = T x Y .

(9.9)

Here and in the following the letter C denotes a generic positive constant independent of x.

N (T ) X

x1

0

x = x0 + x1

x0

D(T ) ∩ X0

Fig. 9.1. The topological decomposition of N (T ) in the domain D(T )

On the other hand, it should be noticed that all norms on a finite dimensional linear space are equivalent. This gives that

286

9 Proof of Theorem 1.4

x1 X ≤ C x1 Z .

(9.10)

Moreover, since the injection X → Z is compact and hence is continuous, we obtain that

x1 Z ≤ x Z + x0 Z ≤ x Z + C x0 X . (9.11) Thus we have, by inequalities (9.10) and (9.11),

x1 X ≤ C ( x Z + x0 X ) .

(9.12)

Therefore, by combining inequalities (9.9) and (9.12) we obtain the desired inequality

x X ≤ x0 X + x1 X ≤ C ( T x Y + x Z )

for all x ∈ D(T ),

(9.7)

since T x0 = T x. (ii) =⇒ (i): Conversely, by inequality (9.7) it follows that

x X ≤ C x Z

for all x ∈ N (T ).

(9.13)

However, the null space N (T ) is closed in X, and so it is a Banach space. Since the injection X → Z is compact, we obtain from inequality (9.13) that the closed unit ball {x ∈ N (T ) : x X ≤ 1} of the Banach space N (T ) is compact. Hence it follows from an application of [147, Chapter III, Section 2, Corollary 2] that dim N (T ) < ∞. Let X0 be a closed topological complement of N (T ) as in the topological decomposition (9.8). To prove the closedness of R(T ), it suffices to show that

x X ≤ C T x Y

for all x ∈ D(T ) ∩ X0 .

The proof is based on a reduction to absurdity. Assume, to the contrary, that: For every n ∈ N, there is an element xn of D(T ) ∩ X0 such that

xn X > n T xn Y . If we let

xn :=

xn ,

xn X

then we have the assertions xn ∈ D(T ) ∩ X0 ,

xn X = 1,

(9.14a)

9.1 Proof of Theorem 1.4, Part (i)

T xn Y
0, the operator Ap − με I satisfies conditions (2.1) and (2.2) (see Figure 9.2 below). Thus, by applying Theorem 2.2 (and Remark 2.3) to the operator Ap −με I we obtain part (ii) of Theorem 1.4: Theorem 9.11. Assume that conditions (A) and (B) are satisfied. Then the operator Ap generates a semigroup Uz on Lp (D) which is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2, and enjoys the following three properties (a), (b) and (c) : p z (a) The operators Ap Uz and dU dz are bounded operators on L (D) for each z ∈ Δε , and satisfy the relation

dUz = Ap Uz dz

for all z ∈ Δε .

9.3 Notes and Comments

293

Σω

Fig. 9.2. The operator Ap − με satisfies conditions (2.1) and (2.2)

!0 (ε), M !1 (ε) and με (b) For each 0 < ε < π/2, there exist positive constants M such that !0 (ε)eμε ·Re z

Uz ≤ M

for all z ∈ Δε ,

!1 (ε) M eμε ·Re z

Ap Uz ≤ |z|

for all z ∈ Δε .

(c) For each u0 ∈ Lp (D), we have, as z → 0, z ∈ Δε , Uz u0 −→ u0

in Lp (D).

The proof of Theorem 1.4 is now complete.  

9.3 Notes and Comments This chapter is a revised and expanded version of Chapter 7 of the second edition [121]. The material is based on the lecture entitled Spectral Analysis of the Subelliptic Oblique Derivative Problem delivered at Partial Differential Equations and Applications, Universit` a di Bologna, Bologna, Italy, on May 24, 2017. Section 9.1: For the detailed proof of Proposition 9.2, the reader might be referred to [113] and [114, Section 8.3].

Part IV

Waldenfels Operators, Boundary Operators and Maximum Principles

10 Elliptic Waldenfels Operators and Maximum Principles

Part IV (Chapters 10 and 11) is devoted to the general study of the maximum principles for second-order, elliptic Waldenfels operators in terms of pseudodifferential operators. In this chapter, following Bony–Courr`ege–Priouret [15] we prove various maximum principles for second-order, elliptic Waldenfels operators which play an essential role throughout the book. In Section 10.1 we give complete characterizations of linear operators W = A + S which satisfy the positive maximum principle (PM) closely related to condition (β  ) given in the Hille–Yosida–Ray theorem (Theorem 3.36): ◦

x0 ∈ D, v ∈ C02 (D) and v(x0 ) = sup v(x) ≥ 0 =⇒ W v(x0 ) ≤ 0

(PM)

x∈D

(Theorems 10.1, 10.2 and 10.4). In Section 10.2 we prove the weak and strong maximum principles and Hopf’s boundary point lemma for second-order, elliptic Waldenfels operators W = A + S (Theorems 10.5, 10.7 and Lemma 10.11) that play an important role in Part V.

10.1 Borel Kernels and Maximum Principles Let D be an open subset of Euclidean space RN or of the half-space RN +. N More precisely, if D is an open set in RN in the topology induced on R + + N from R , then we define the boundary ∂D of D to be the intersection of D ◦

with RN −1 × {0} and the interior D of D to be the complement of ∂D in D, that is,   ∂D = D ∩ x ∈ RN : xN = 0 , ◦   D = D ∩ x ∈ RN : xN > 0 .

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 10

298

10 Elliptic Waldenfels Operators and Maximum Principles

We remark (see Figures 10.1 and 10.2 below) that ⎧◦ ⎨D if D is open in RN , D= ◦ ⎩D ∪ ∂D if D is open in RN . +

xN

◦

D=D x = (x1 , . . . , xN −1 )

0

Fig. 10.1. D is open in RN

xN ◦

D = D ∪ ∂D ◦

D 0

∂D

x = (x1 , . . . , xN −1 )

Fig. 10.2. D is open in RN +

Then we let ◦

◦

Bloc (D) = the space of Borel measurable functions in D ◦

which are bounded on compact subsets of D, and C0 (D) = the space of continuous functions in D with compact support in D.

10.1 Borel Kernels and Maximum Principles

299

Let BD and B ◦ be the σ-algebra of all Borel sets in D and the σ-algebra D

◦

of all Borel sets in D, respectively. ◦

A positive Borel kernel of D into D is a mapping ◦

D  x −→ s(x, dy) ◦

of D into the space of non-negative measures on BD such that, for each X ∈ BD the function  ◦ D  x −→ s(x, X) = s(x, dy) (10.1) X ◦

is Borel measurable in D. A local unity function on D is a smooth function σ(x, y) in D × D which satisfies the following three conditions (LU1), (LU2) and (LU3) (see Figure 10.3 below): (LU1) 0 ≤ σ(x, y) ≤ 1 in D × D. (LU2) σ(x, y) = 1 in a neighborhood of the diagonal ΔD = {(x, x) : x ∈ D} in the product space D × D. (LU3) For any compact subset K of D, there exists a compact subset K  of D such that the functions σx (·) = σ(x, ·) for all x ∈ K have their support in K  . supp σ

D

K

D K

Fig. 10.3. The condition (LU3)

We can construct a local unity function σ(x, y) in the following way (see [41, Proposition (8.15)]): Let {Ui }i∈I be a locally finite open covering of D and let {ϕi }i∈I be a partition of unity subordinate to the covering {Ui }. Namely, the family {ϕi }i∈I satisfies the following three conditions (PU1), (PU2) and (PU3):

300

10 Elliptic Waldenfels Operators and Maximum Principles

(PU1) 0 ≤ ϕi (x) ≤ 1 for all x ∈ D and i ∈ I. (PU2) supp ϕi ⊂ Ui for each i ∈ I. (PU3) The collection {supp ϕi }i∈I is locally finite and  ϕi (x) = 1 for every x ∈ D. i∈I

Here supp ϕi is the support of ϕi , that is, the closure in D of the set {x ∈ D : ϕi (x) = 0}. If we take a smooth function ψi (x) in D such that  0 ≤ ψi (x) ≤ 1 for all x ∈ D, ψi (x) = 1 on supp ϕi , then it is easy to verify that the function  σ(x, y) = ϕi (x) ψi (y) for (x, y) ∈ D × D, i∈I

satisfies the desired conditions (LU1), (LU2) and (LU3). 10.1.1 Linear Operators having Positive Borel Kernel Now we assume that a positive Borel kernel s(x, dy) satisfies the following two conditions (NS.1) and (NS.2): ◦

(NS.1) s(x, {x}) = 0 for every x ∈ D. (NS.2) For every non-negative function f (x) in C0 (D), the function  ◦ D  x −→ s(x, dy)|y − x|2 f (y) D ◦

belongs to the space Bloc (D) of locally bounded, Borel measurable ◦

functions on D. By using Taylor’s formula and condition (NS.2), we can define a linear operator ◦

S : C02 (D) −→ Bloc (D) by the formula (see Example 10.1) Su(x) (10.2) 9 $ %:  N  ∂u = s(x, dy) u(y) − σ(x, y) u(x) + (x)(yi − xi ) ∂xi D i=1

10.1 Borel Kernels and Maximum Principles ◦

for every x ∈ D

301

(u ∈ C02 (D)).

Here C02 (D) is the space of functions in C 2 (D) with compact support in D. (I) First, we give a complete characterization of linear continuous operators ◦

W : C02 (D) −→ Bloc (D) that have positive Borel kernels in the case where D is an open subset of RN or of RN eor`eme I], [117, Theorem 8.2]): + (see [15, Th´ Theorem 10.1. Let D be an open subset of RN or of RN + . If W is a linear ◦

operator from C02 (D) into Bloc (D), then the following two assertions (p0 ) and (w) are equivalent: (p0 ) The operator ◦

W : C02 (D) −→ Bloc (D) is continuous and satisfies the condition ◦

x ∈ D, u ∈ C02 (D), u ≥ 0 in D and x ∈ supp u =⇒ W u(x) ≥ 0.

(10.3)

(w) There exist a second-order, differential operator ◦

A : C 2 (D) −→ Bloc (D) and positive Borel kernels s(x, dy), having properties (NS.1) and (NS.2), such that the operator W is written in the form W u(x) = Au(x) + Su(x) ⎛ ⎞ N N 2   ∂ u ∂u =⎝ aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ ∂x ∂x ∂x i j i i,j=1 i=1

(10.4)

   N  ∂u + s(x, dy) u(y) − σ(x, y) u(x) + (x)(yi − xi ) , ∂xi D i=1 

◦

for every x ∈ D

(u ∈ C02 (D)). ◦

Here the coefficients aij (x), bi (x) and c(x) belong to the space Bloc (D). (II) Secondly, we give a useful characterization of linear continuous operators ◦ W : C02 (D) −→ Bloc (D) that have positive Borel kernels in the case where D is an open subset of RN or of RN eor`eme II], [117, Theorem 8.4]): + (see [15, Th´

302

10 Elliptic Waldenfels Operators and Maximum Principles

Theorem 10.2. Let D be an open subset of RN or of RN + . Let V be a linear subspace of C02 (D) that contains C0∞ (D): C0∞ (D) ⊂ V ⊂ C02 (D). ◦

Assume that W is a linear operator from V into Bloc (D) and satisfies the condition ◦

x ∈ D, u ∈ V, u ≥ 0 in D and u(x) = 0 =⇒ W u(x) ≥ 0.

(10.5)

Then the operator W can be extended uniquely to a linear operator from C02 (D)

◦

into Bloc (D) in such a way that:

(p1 ) The extended operator ◦

W : C02 (D) −→ Bloc (D) is continuous and satisfies condition (10.5) for all u ∈ C02 (D): ◦

x ∈ D, u ∈ C02 (D), u ≥ 0 in D and u(x) = 0 =⇒ W u(x) ≥ 0. (10.5 ) By combining Theorems 10.1 and 10.2, we have the following simple characterization of W (see [117, Corollary 8.7]): Corollary 10.3. Let D be an open subset of RN or of RN + . If a linear operator ◦

W : C02 (D) −→ Bloc (D) satisfies condition (p1 ), then it is continuous and can be written in the form (10.4) W = A + S, ◦

where the coefficients aij (x), bi (x) and c(x) of A belong to the space Bloc (D) and the positive Borel kernels s(x, dy) of S enjoy properties (NS.1) and (NS.2). ◦

Proof. If W : C02 (D) → Bloc (D) is a linear operator, then it follows from an application of Theorem 4.2 that condition (p1 ) implies the continuity of W and condition (p0 ). Therefore, we obtain from Theorem 4.1 that condition (w) is satisfied. The proof of Corollary 10.3 is complete.   We remark that if the integro-differential operator W =A+S

10.1 Borel Kernels and Maximum Principles

303

enjoys property (w), then the Borel kernel s(x, dy) and the principal part

 ij  a (x) 1≤i,j≤n

of A are uniquely determined by W . Indeed, it suffices to note the following two formulas (10.6) and (10.7):  ◦ s(x, dy) u(y) = W u(x) if u ∈ C02 (D) and x ∈ D \ supp u. (10.6) D

2 u(x)

N 

  aij (x)ξi ξj = W Φξx u (x) −

i,j=1

 D

s(x, dy) Φξx (y)u(y)

(10.7)

◦

if x ∈ D, u ∈ C02 (D) and ξ = (ξi ) ∈ RN , where

$ Φξx (y)

=

N 

%2 ξi (yi − xi )

for x, y ∈ D and ξ = (ξi ) ∈ RN .

i=1

(III) Finally, we characterize a class of linear operators ◦

W = A + S : C02 (D) −→ Bloc (D) that satisfy the positive maximum principle (PM) (see [15, Th´eor`eme III], [117, Theorem 8.8]): Theorem 10.4. Let D be an open subset of RN or of RN + . If W is a lin◦

ear operator from C02 (D) into Bloc (D) of the form (10.4), then we have the following two assertions (i) and (ii): (i) The operator W satisfies condition (10.5) if and only if the principal sym◦ &N bol − i,j=1 aij (x)ξi ξj of A is non-positive on D × RN . (ii) The operator W satisfies the positive maximum principle (PM) ◦

x0 ∈ D, v ∈ C02 (D) and v(x0 ) = sup v(x) ≥ 0 =⇒ W v(x0 ) ≤ 0 (PM) x∈D

◦

if and only if the principal symbol of A is non-positive on D × RN and ◦

the following two conditions (10.8a) and (10.8b) hold true for all x ∈ D: (A1)(x) = c(x) ≤ 0,  (W 1)(x) = c(x) + s(x, dy)[1 − σ(x, y)] ≤ 0. D

(10.8a) (10.8b)

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10 Elliptic Waldenfels Operators and Maximum Principles

In particular, the positive Borel kernels s(x, dy) enjoy the following property (NS.3): ◦

(NS.3) For any open subset D of D, the function  D  x −→ s(x, D \ D ) =

s(x, dy)

D\D

belongs to the space Bloc (D ). 10.1.2 Borel Kernels and Pseudo-Differential Operators Let D be a bounded domain of Euclidean space RN with smooth boundary ∂D. In this subsection we give two important examples of positive Borel kernels in terms of pseudo-differential operators: Example 10.1. Let s(x, y) be the distribution kernel of a properly supported, N pseudo-differential operator S ∈ L2−κ 1,0 (R ) with κ > 0, and s(x, y) ≥ 0 off the N diagonal ΔRN = {(x, x) : x ∈ R } in the product space RN × RN . Then the integro-differential operator Sr , defined by the formula (see formula (10.2)) Sr u(x) ⎡ ⎛ ⎞⎤  N  ∂u = s(x, y) ⎣u(y) − σ(x, y) ⎝u(x) + (yj − xj ) (x)⎠⎦ dy, ∂x j D j=1 is absolutely convergent for every x ∈ D. Proof. Indeed, we can write the integral Sr u(x) in the form Sr u(x)  = s(x, y) [1 − σ(x, y)] u(y) dy D ⎛ ⎞  N  ∂u s(x, y)σ(x, y) ⎝u(y) − u(x) − (yj − xj ) (x)⎠ dy. + ∂x j D j=1 Then, by using Taylor’s formula u(y) − u(x) −

N 

(yj − xj )

j=1

=

N  i,j=1

(yi − xi )(yj − xj )

∂u (x) ∂xj

 0

1

(1 − t)

 ∂2u (x + t(y − x))dt , ∂xi ∂xj

10.1 Borel Kernels and Maximum Principles

we can find a constant C1 > 0 such that     N    ∂u u(y) − u(x) −  ≤ C1 |x − y|2 (y − x ) (x) j j   ∂x j   j=1

305

for all x, y ∈ D.

On the other hand, by using Theorem 4.51 with n := N we can prove that N the distribution kernel s(x, y) of a pseudo-differential operator S ∈ Lm 1,0 (R ) satisfies the estimate |s(x, y)| ≤

C |x − y|m+N

for all x, y ∈ RN and x = y.

Therefore, by taking the compact set D ⊂ RN and m := 2 − κ we can find N a constant C2 > 0 such that the distribution kernel s(x, y) of S ∈ L2−κ 1,0 (R ) satisfies the estimate 0 ≤ s(x, y) ≤

C2 |x − y|N +2−κ

for all x, y ∈ D and x = y.

Hence we have the estimate  ⎛ ⎞    N    ∂u  s(x, y)σ(x, y) ⎝u(y) − u(x) − ⎠ (yj − xj ) (x) dy   ∂xj  D  j=1  1 · |x − y|2 dy ≤ C1 C2 u C 2 (D) N +2−κ D |x − y|  1 dy. = C1 C2 u C 2 (D) N −κ |x − y| D Similarly, we have, with some constant C3 > 0,       s(x, y) [1 − σ(x, y)] u(y) dy  ≤ C3 u C(D)   D

D

1 dy, |x − y|N −κ

since we have the formula σ(x, y) − 1 = σ(x, y) − σ(x, x) −

N 

(yj − xj )

j=1

=

N  i,j=1



(yi − xi )(yj − xj )

0

1

∂σ (x, x) ∂xj

(1 − t)

 ∂2σ (x, x + t(y − x))dt . ∂xi ∂xj

Therefore, we obtain that the integral Sr u(x) is absolutely convergent. The proof of Example 10.1 is complete.  

306

10 Elliptic Waldenfels Operators and Maximum Principles

Example 10.2. Let r(x , y  ) be the distribution kernel of a pseudo-differential   1 operator R ∈ L2−κ 1,0 (∂D) with κ1 > 0, and r(x , y ) ≥ 0 off the diagonal    Δ∂D = {(x , x ) : x ∈ ∂D} in the product space ∂D × ∂D. Let t(x, y) be the distribution kernel of a properly supported, pseudo-differential operator N 2 T ∈ L1−κ 1,0 (R ) with κ2 > 0, and t(x, y) ≥ 0 off the diagonal ΔRN = {(x, x) : N x ∈ R } in the product space RN ×RN . Then the integro-differential operator Γr , defined by the formula Γr u(x ) ⎡ ⎛ ⎞⎤  N −1  ∂u = r(x , y  ) ⎣u(y  ) − τ (x , y  ) ⎝u(x ) + (yj − xj ) (x )⎠⎦ dy  ∂x j ∂D j=1  + t(x , y) [u(y) − u(x )] dy, D

is absolutely convergent for every x ∈ ∂D. 1−κ2 N 1 Proof. Since R ∈ L2−κ 1,0 (∂D) and T ∈ L1,0 (R ), it follows from an application of [122, Section 7.8, Theorem 7.36] with n := N that the kernels r(x , y  ) and t(x , y) satisfy respectively the estimates

0 ≤ r(x , y  ) ≤

C

for all x , y  ∈ ∂D and x = y  , |x − y  |(N −1)+2−κ1 C  0 ≤ t(x , y) ≤  for all x ∈ ∂D and y ∈ D, |x − y|(N −1)+2−κ2

where |x − y  | denotes the geodesic distance between x and y  with respect to the Riemannian metric of ∂D. Therefore, by arguing just as in Example 10.1 we find that the integrals Rr u(x ) ⎡ ⎛ ⎞⎤  N −1  ∂u  ⎠⎦  r(x , y  ) ⎣u(y  ) − τ (x , y  ) ⎝u(x ) + (yj − xj ) (x ) dy = ∂x j ∂D j=1 and Tr u(x ) =



t(x , y) [u(y) − u(x )] dy

D

are both absolutely convergent for every x ∈ ∂D. The proof of Example 10.2 is complete.  

10.2 Maximum Principles for Elliptic Waldenfels Operators

307

10.2 Maximum Principles for Elliptic Waldenfels Operators In this section we prove the weak and strong maximum principles and Hopf’s boundary point lemma for second-order, elliptic Waldenfels integro-differential operators which play an essential role in Chapters 12 and 13. Let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D. We consider a second-order, elliptic Waldenfels operator W with real coefficients such that W u(x) = Au(x) + Su(x) N 

(1.2) N 

∂2u ∂u (x) + bi (x) (x) + c(x)u(x) ∂x ∂x ∂x i j i i,j=1 i=1 ⎛ ⎞  N  ∂u ⎝u(x + z) − u(x) − + zj (x)⎠ s(x, z) m(dz). ∂x j RN \{0} j=1

:=

aij (x)

Here: (1) aij (x) ∈ C ∞ (D), aij (x) = aji (x) for all x ∈ D and 1 ≤ i, j ≤ N , and there exists a constant a0 > 0 such that N 

aij (x)ξi ξj ≥ a0 |ξ|2

for all (x, ξ) ∈ D × RN .

i,j=1

(2) bi (x) ∈ C ∞ (D) for all 1 ≤ i ≤ N . (3) c(x) ∈ C ∞ (D), and c(x) ≤ 0 in D, but c(x) ≡ 0 in D. (4) s(x, z) ∈ L∞ (RN × RN ) and 0 ≤ s(x, z) ≤ 1 almost everywhere in RN × RN , and there exist constants C0 > 0 and 0 < θ < 1 such that |s(x, z) − s(y, z)| ≤ C0 |x − y|θ

(1.3a)

for all x, y ∈ D and almost all z ∈ R , N

and s(x, z) = 0

if x ∈ D and x + z ∈ D.

(1.3b)

Probabilistically, condition (1.3b) implies that all jumps from D are within D. Analytically, condition (1.3b) guarantees that the integral operator S may be considered as an operator acting on functions u defined on the closure D (see Garroni–Menaldi [48, Chapter II, Remark 1.19]). (5) The measure m(dz) is a Radon measure on RN \ {0} which has a density with respect to the Lebesgue measure dz on RN , and satisfies the moment condition (see Example 1.1)   |z|2 m(dz) + |z| m(dz) < ∞. (1.4) {01}

308

10 Elliptic Waldenfels Operators and Maximum Principles

The moment condition implies that the measure m(·) admits a singularity of order 2 at the origin, and this singularity at the origin is produced by the accumulation of small jumps of Markovian particles, while the measure m(·) admits a singularity of order 1 at infinity, and this singularity at infinity is produced by the accumulation of large jumps of Markovian particles. Finally, it should be noticed that (W 1)(x) = (A1)(x) + (S1)(x) = c(x) ≤ 0

in D.

(1.5)

10.2.1 Weak Maximum Principle First, we prove the weak maximum principle for elliptic, Waldenfels integrodifferential operators (see [136, p. 191, Lemma 3.25], [122, Theorem 8.11], [127, Theorem 8.1]): Theorem 10.5 (the weak maximum principle). Let W be a second-order, elliptic Waldenfels operator. Then we have the following two assertions (i) and (ii): (i) If a function u(x) ∈ C(D) ∩ C 2 (D) satisfies the conditions  W u(x) ≥ 0 in D, W 1(x) < 0 in D, then the function u(x) may take its positive maximum only on the boundary ∂D. (ii) If a function u(x) ∈ C(D) ∩ C 2 (D) satisfies the conditions  W u(x) > 0 in D, W 1(x) ≤ 0 in D, then the function u(x) may take its non-negative maximum only on the boundary ∂D. Proof. Our proof is based on a reduction to absurdity. Assume, to the contrary, that there exists a point x0 of D such that u(x0 ) = max u(x). x∈D

Then we have the assertions ∂u (x0 ) = 0 for 1 ≤ i ≤ N , ∂xi N  ∂2u • aij (x0 ) (x0 ) ≤ 0, ∂xi ∂xj i,j=1 •

10.2 Maximum Principles for Elliptic Waldenfels Operators

309

and hence the inequalities • Au(x0 ) =

N 

aij (x0 )

i,j=1

 • Su(x0 ) =

RN \{0}

∂2u (x0 ) + c(x0 )u(x0 ) ≤ c(x0 )u(x0 ), ∂xi ∂xj

(u(x0 + z) − u(x0 )) s(x0 , z) m(dz) ≤ 0.

(10.9) (10.10)

Assertion (i): If W u(x) ≥ 0 in D, c(x) = W 1(x) < 0 in D and if u(x0 ) = maxD u > 0, then it follows from inequalities (10.9) and (10.10) that 0 ≤ W u(x0 ) = Au(x0 ) + Su(x0 ) ≤ max u(x) · W 1(x0 ) < 0. x∈D

This is a contradiction. Assertion (ii): Similarly, if W u(x) > 0, c(x) = W 1(x) ≤ 0 in D and if u(x0 ) = maxD u ≥ 0, then it follows from inequalities (10.9) and (10.10) that 0 < W u(x0 ) = Au(x0 ) + Su(x0 ) ≤ max u(x) · W 1(x0 ) ≤ 0. x∈D

This is also a contradiction. The proof of Theorem 10.5 is complete.   As an application of the weak maximum principle, we can obtain a pointwise estimate for solutions of the non-homogeneous equation W u = f : Theorem 10.6. Let W be a second-order, elliptic Waldenfels operator. Assume that W 1(x) < 0 on D = D ∪ ∂D. Then we have, for all u ∈ C(D) ∩ C 2 (D),    1 max |u| ≤ max sup |W u|, max |u| . ∂D minD (−W 1) D D

(10.11)

Proof. We let  M = max

1 minD (−W 1)



 sup |W u|, max |u| , D

∂D

and consider two functions v± (x) = M ± u(x). Then it follows that W v± (x) = M · W 1(x) ± W u(x) ≤ 0 in D. Hence, by applying part (i) of Theorem 10.5 to the functions −v± (x) we obtain that the functions v± (x) may take their negative minimums only on the boundary ∂D. However, it follows that

310

10 Elliptic Waldenfels Operators and Maximum Principles

v± (x) = M ± u(x) ≥ 0 on the boundary ∂D. Hence we have the inequality v± (x) ≥ 0 on D. This proves the desired estimate (10.11). The proof of Theorem 10.6 is complete.   10.2.2 Strong Maximum Principle The next theorem is a generalization of the strong maximum principle for the Laplacian to the integro-differential operator case (see [15, Th´eor`eme VII], [136, p. 193, Theorem 3.27], [122, Theorem 8.13], [128, Theorem 6.2]): Theorem 10.7 (the strong maximum principle). Let W be a second order, elliptic Waldenfels operator. Assume that a function u(x) ∈ C 2 (D) satisfies the conditions  W u(x) ≥ 0 in D, maxx∈D u(x) ≥ 0. If the function u(x) takes its non-negative maximum at an interior point of D, then it is a constant. Proof. The proof is divided into four steps. Step (I): Our proof is based on a reduction to absurdity. First, we let M = max u(x) ≥ 0, x∈D

F = {x ∈ D : u(x) = M } , and assume, to the contrary, that F  D. Since F is closed in D, we can find a point x0 of F and an open ball V contained in the set D \ F , centered at x1 , such that (see Figure 10.4 below) (a) V ⊂ D \ F ; (b) x0 is on the boundary ∂V of V . Step (II): Secondly, we study the integral operator S in the framework of H¨older spaces. To do this, we need the following elementary estimates (10.12a), (10.12b) and (10.12c) for the measure m(dz): Claim 10.8. For each ε > 0, we let  σ(ε) = {0ε}

|z| m(dz),



m(dz).

τ (ε) = {|z|>ε}

Then we have, as ε ↓ 0, σ(ε) −→ 0, C1 + C2 , δ(ε) ≤ ε C1 τ (ε) ≤ 2 + C2 , ε where

 C1 =

{01}

|z| m(dz).

Proof. Assertion (10.12a) follows immediately from condition (1.4). The term δ(ε) can be estimated as follows:   δ(ε) = |z| m(dz) + |z| m(dz) {|z|>1} {ε1} {ε1} {0 0 in V . v(x) < 0 on D \ V . v(x) = 0 on ∂V . W v(x0 ) > 0.

Remark 10.10. In Lemma 13.3, we shall prove the following assertion: v ∈ C 2+θ0 (D) =⇒ Sv ∈ C θ0 (D) =⇒ W v = Av + Sv ∈ C θ0 (D). Proof. In order to prove Claim 10.9, we define a function v(x) by the formula     v(x) = exp −q|x − x1 |2 − exp −qρ2 , ρ = |x0 − x1 |, where q is a positive constant to be chosen later on. Then it is easy to see that the function v(x) satisfies conditions (i) through (iii). Hence it suffices to show that v(x) satisfies condition (iv) for q sufficiently large. (1) First, we estimate the function Av(x0 ) from below. To do this, it should be noticed that v(x0 ) = 0,

  ∇v(x0 ) = 2q(x1 − x0 ) exp −qρ2 = 0.

(10.13a) (10.13b)

Hence we have the formula  N  Av(x0 ) = 4q 2 aij (x0 )(xi1 − xi0 )(xj1 − xj0 ) i,j=1

− 2q

N  

ii

a (x0 ) + b

i

(x0 )(xi0

−

' 

xi1 )

  exp −qρ2 .

i=1

Since the matrix (aij ) is positive definite, we can estimate the function Av(x0 ) from below as follows:     (10.14) Av(x0 ) ≥ 4a0 ρ2 q 2 − C q exp −qρ2 , where C > 0 is a constant independent of q. (2) Secondly, in order to estimate the function Sv(x0 ) we study the kernel s(x0 , z). We recall that u(x0 ) = M = max u(x) ≥ 0. x∈D

Hence we have the inequalities

10.2 Maximum Principles for Elliptic Waldenfels Operators N 

Au(x0 ) =

aij (x0 )

i,j=1

313

∂2u (x0 ) + c(x0 )u(x0 ) ≤ c(x0 )u(x0 ) ≤ 0, ∂xi ∂xj



and Su(x0 ) =

RN \{0}

s(x0 , z) (u(x0 + z) − u(x0 )) m(dz) ≤ 0.

This implies that Au(x0 ) = 0, Su(x0 ) = 0. Indeed, it suffices to note that 0 ≤ W u(x0 ) = Au(x0 ) + Su(x0 ) ≤ 0. Thus we obtain that

 0 = Su(x0 ) = s(x0 , z) (u(x0 + z) − u(x0 )) m(dz) RN \{0}  = s(x0 , z) (u(x0 + z) − u(x0 )) m(dz), x0 +z ∈F

so that s(x0 , z) = 0

if x0 + z ∈ F ,

since we have, for all x0 + z ∈ F , u(x0 + z) − u(x0 ) < 0. Therefore, we can write the function Sv(x0 ) in the form (see Figure 10.5 below) Sv(x0 )  =

x0 +z∈F z =0

⎛

⎞ ∂v s(x0 , z) ⎝v(x0 + z) − v(x0 ) − zj (x0 )⎠ m(dz). ∂x j j=1 N 

For each ε > 0, we decompose the function Sv(x0 ) into the two terms Sv(x0 )  =

x0 +z∈F 0ε

⎞

∂v (x0 )⎠ m(dz) ∂x j j=1 ⎛ ⎞ N  ∂v s(x0 , z) ⎝v(x0 + z) − v(x0 ) − zj (x0 )⎠ m(dz) ∂x j j=1

s(x0 , z) ⎝v(x0 + z) − v(x0 ) −

 +

N 

zj

314

10 Elliptic Waldenfels Operators and Maximum Principles

x1 x0

V

x0 + z

F

Fig. 10.5. s(x0 , z) for x0 + z ∈ F (ε)

(ε)

:= S1 v(x0 ) + S2 v(x0 ). Then, by using formulas (10.13) and Claim 10.8 we can estimate the second (ε) term S2 v(x0 ) as follows:    (ε)  (10.15) S2 v(x0 )      N    ∂v  ≤ s(x0 , z) v(x0 + z) − v(x0 ) − zj (x0 ) m(dz) x0 +z∈F ∂xj   j=1 |z|>ε  '    ≤ m(dz) + 2qρ |z| m(dz) exp −qρ2 |z|>ε

|z|>ε

  = {τ (ε) + 2qρδ(ε)} exp −qρ2       C1 C1 exp −qρ2 . ≤ + C2 + 2qρ + C2 2 ε ε Indeed, it suffices to note (see Figure 10.5) that |x0 + z − x1 | ≥ |x0 − x1 | = ρ

for all x0 + z ∈ F ,

so that   exp −q|x + z0 − x1 |2     ≤ exp −q|x0 − x1 |2 = exp −qρ2

for all x0 + z ∈ F . (ε)

Similarly, we can estimate the first term S1 v(x0 ) as follows:    (ε)  (10.16) S1 v(x0 )      N    ∂v ≤ s(x0 , z) v(x0 + z) − v(x0 ) − zj (x0 ) m(dz) x0 +z∈F ∂xj   j=1 0 0 is a constant independent of q. (2) Secondly, in order to estimate the function Sv(x0 ) we study the kernel s(x0 , z): By conditions (10.18) and (10.19), it follows that ∂u  (x ) = 0 for 1 ≤ i ≤ N , ∂xi 0 ∂2u  (x ) ≤ 0. • ∂x2N 0

•

Hence we have the inequality Au(x0 ) =

N  i,j=1

aij (x0 )

∂2u (x ) + c(x0 )u(x0 ) ∂xi ∂xj 0

320

10 Elliptic Waldenfels Operators and Maximum Principles

= aN N (x0 )

N −1  ∂2u  ∂2u ij  (x ) + a (x ) (x0 ) + c(x0 )u(x0 ) 0 ∂x2N 0 ∂x ∂x i j i,j=1

≤ 0, and also Su(x0 ) =

 RN \{0}

s(x0 , z) (u(x0 + z) − u(x0 )) m(dz) ≤ 0.

This implies that Au(x0 ) = 0, Su(x0 ) = 0. Indeed, it suffices to note that 0 ≤ W u(x0 ) = Au(x0 ) + Su(x0 ) ≤ 0. Thus we obtain that 0 = Su(x0 )  = s(x0 , z) (u(x0 + z) − u(x0 )) m(dz) RN \{0}  = s(x0 , z) (u(x0 + z) − u(x0 )) m(dz), x0 +z ∈G

so that

s(x0 , z) = 0 if x0 + z ∈ ∂D \ G,

since we have, for all x0 + z ∈ ∂D \ G, u(x0 + z) − u(x0 ) < 0. 

Here G=

 x ∈ ∂D : u(x ) = max u(x) . x∈D

Moreover, it should be noticed that x0 + z ∈ G ⇐⇒ z = (z  , 0) ∈ G. Therefore, we can write the function Sv(x0 ) in the form Sv(x0 )  = 

⎞ ∂v s(x0 , z) ⎝v(x0 + z) − v(x0 ) − zj (x0 )⎠ m(dz). x0 +z∈G ∂x j j=1 z =0

⎛

N 

10.2 Maximum Principles for Elliptic Waldenfels Operators

321

For each ε > 0, we decompose the function Sv(x0 ) into the two terms Sv(x0 )  = 

⎛

⎞ ∂v s(x0 , z) ⎝v(x0 + z) − v(x0 ) − zj (x0 )⎠ m(dz) x0 +z∈G ∂x j j=1 0ε

:=

(ε) S1 v(x0 )

(ε)

+ S2 v(x0 ). (ε)

By using formulas (10.20), we can estimate the second term S2 v(x0 ) as follows:    (ε)  (10.22) S2 v(x0 )        ≤ s(x0 , z) exp −q(|z  |2 + (zN − ρ)2 ) − exp −qρ2 x +z∈G 0

|z|>ε, zN =0

  − 2ρzN q exp −qρ2 m(dz)    ≤ m(dz) exp −qρ2 |z|>ε zN =0

  ≤ τ (ε) exp −qρ2     C1 ≤ + C2 exp −qρ2 . 2 ε (ε)

Similarly, we can estimate the first term S1 v(x0 ) as follows:    (ε)  S1 v(x0 )  s(x0 , z)v(x0 + z) m(dz) ≤ 

(10.23)

x0 +z∈G 0 0 in V  ∩ D. (c) uλ = u + λv ≤ u ≤ M on D \ V , since v ≤ 0 on D \ V . (d) uλ = u + λv ≤ M on V \ V  for λ sufficiently small, since u < M on V \ V  .

xN V V

∂D

x0

x

Fig. 10.11. The open ball V and the neighborhood V  of x0

Hence it follows from an application of part (ii) of Theorem 10.5 that uλ ≤ M

in V  ∩ D,

so that uλ (y) = u(y) + λv(y) ≤ M = u(x0 ) + λv(x0 ),

y ∈ V  ∩ D,

10.3 Notes and Comments

323

or equivalently u(y) − u(x0 ) ≤ −λ(v(y) − v(x0 )),

y ∈ V  ∩ D.

Therefore, we obtain that ∂u  ∂v (x ) ≤ −λ (x0 ) < 0. ∂n 0 ∂n This contradicts hypothesis (10.19). Now the proof of Lemma 10.11 is complete.  

10.3 Notes and Comments This chapter is a modern version of the Bony–Courr`ege–Priouret theory ([15]) from the viewpoint of pseudo-differential operators with the special emphasis on the transmission property introduced by Boutet de Monvel [19]. Protter–Weinberger [91] is the classic for maximum principles. For a general study of maximum principles for Waldenfels integro-differential operators, the reader might refer to Bony–Courr`ege–Priouret [15], Troianiello [136, Section 3.7.2], Taira [122, Chapter 8], [127] and [128]. See also Taira [125] for a strong maximum principle for globally hypoelliptic operators. This chapter is based on the lecture entitled A class of hypoelliptic Vishik– Wentzell boundary vale problems delivered at Functional Analysis Methods for Partial Differential Equations, Centro Polifunzionale, Cesena, Italy, on June 27, 2019.

11 Boundary Operators and Boundary Maximum Principles

Let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary (see Figure 11.1 below). In this chapter, following Bony– Courr`ege–Priouret ([15, Chapter II]) we characterize Ventcel’–L´evy boundary operators T (Theorem 11.3) and Ventcel’ boundary operators Γ = Λ + T (Theorem 11.4) defined on the compact smooth manifold D with boundary ∂D in terms of the positive boundary maximum principle: x0 ∈ ∂D, u ∈ C 2 (D) and u(x0 ) = max u(x) ≥ 0 =⇒ (Γ u)(x0 ) ≤ 0. x∈D

(PMB) This chapter will be very useful in the study of Markov processes with general Ventcel’ boundary conditions in the last Chapter 16.

D

∂D Fig. 11.1. The compact smooth manifold D = D ∪ ∂D with boundary ∂D

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 11

326

11 Boundary Operators and Boundary Maximum Principles

11.1 Ventcel’–L´ evy Boundary Operators Let {(Uβ , χβ )}m β=1 be a finite open covering of M = D by local charts. We ∞ choose a family {τβ }m β=1 of functions in C (D × D) such that supp τβ ⊂ Uβ × Uβ and that τ (x, y) =

m 

τβ (x, y) = 1

β=1

in a neighborhood of the diagonal ΔD = {(x, x) : x ∈ D}. We study a positive Borel kernel t(x , dy) which satisfies the following three conditions (NV.1), (NV.2) and (NV.3): (NV.1) t(x , {x }) = 0 for every x ∈ ∂D. (NV.2) For every local chart (Uβ , χβ ) of D such that Uβ ∩ ∂D = ∅ and for every non-negative function f ∈ C(D) with supp f ⊂ Uβ , the function Uβ ∩ ∂D  x →

 Uβ

N −1   β  2  χj (y) − χβj (x ) t(x , dy) f (y) χN (y) + j=1

belongs to the space Bloc (Uβ ∩ ∂D) of locally bounded, Borel measurable functions on Uβ ∩ ∂D. (NV.3) For every non-negative function f ∈ C(D), the function 



∂D  x −→

  m   t(x , dy) f (y) 1 − τβ (x , y) f (y) 

D

(11.1)

β=1

belongs to the space B(∂D) of bounded, Borel measurable functions on ∂D. The Borel kernel t(x , dy) is called a Ventcel’ kernel on the manifold D. We can give a global equivalent version of the conditions (NV.2) and (NV.3) in the following way: For points x ∈ ∂D and y ∈ D, we introduce a function ⎤ ⎡ m N −1  2   y = 7 x χβj (y) − χβj (x ) ⎦ . τβ (x , y) ⎣χβN (y) + (11.2) β=1

j=1

Then it is easy to see that the conditions (NV.2) and (NV.3) are equivalent to the following global condition (NV.4):

11.1 Ventcel’–L´evy Boundary Operators

327

(NV.4) For every non-negative function f ∈ C(D), the function   y f (y) 7 t(x , dy) x ∂D  x −→ D

belongs to the space B(∂D). Now we are in a position to define Ventcel’–L´evy boundary operators; Definition 11.1. We are given the following: (1) t(x , dy) is a Ventcel’ kernel on the manifold D. (2) ζ(x ) is a bounded, Borel measurable vector field on the boundary ∂D: More precisely, we have, in a local chart (Uβ , χβ ), ζ(x ) · u =

N −1 

ζβj (x )

j=1

∂u ∂χβj

with ζβj ∈ Bloc (Uβ ∩ ∂D).

(3) η(x ) is a bounded, Borel measurable function on the boundary ∂D such that    m     t(x , dy) 1 − τβ (x , y) ≤ 0 on ∂D. (11.3) η(x ) + D

β=1

Then we define a continuous boundary operator T : C 2 (D) −→ B(∂D) by the formula (T u)(x ) = η(x )u(x ) + ζ(x ) · u(x ) +



 t(x , dy) u(y)

(11.4)

D

−

  N −1  ∂u τβ (x , y) u(x ) + (χβj (y) − χβj (x )) β (x ) . ∂χj j=1 β=1 m 

We remark that condition (11.3) is equivalent to the following global one: (T 1)(x ) ≤ 0

on ∂D.

(11.5)

The boundary operator T is called a Ventcel’–L´evy boundary operator. Remark 11.2. In terms of a local chart (U, χ) of D such that U ∩ ∂D = ∅, every Ventcel’–L´evy boundary operator T can be characterized as follows (see [15, p. 440, Lemme]): (V1) For every point x ∈ ∂D, we have the formula   (T u)(x ) = t(x , dy) u(y) whenever x ∈ supp u. D

328

11 Boundary Operators and Boundary Maximum Principles

(V2) For every function u ∈ C 2 (D) with supp u ⊂ U , we have the local expression (T u)(x ) = η(x )u(x ) +

N −1 

ζ j (x )

j=1

∂u  (x ) ∂χj

   N −1  ∂u     + t(x , dy) u(y) − u(x ) + (χj (y) − χj (x )) (x ) ∂χj U j=1 

for every x ∈ U ∩ ∂D. (V3) η(x ) + t(x , D \ U ) ≤ 0 for every x ∈ U ∩ ∂D.

11.2 Positive Boundary Maximum Principles In this section we characterize a general boundary operator of the form Γ = Λ + T : C 2 (D) −→ B(∂D) in terms of the positive boundary maximum principle (see [15, p. 441, Section II.2.6]). 11.2.1 Boundary Maximum Principles for Ventcel’–L´ evy operators First, we prove the following theorem for Ventcel’–L´evy boundary operators T: Theorem 11.3. If T : C 2 (D) → B(∂D) is a Ventcel’–L´evy boundary operator, then T satisfies the positive boundary maximum principle: x0 ∈ ∂D, u ∈ C 2 (D) and u(x0 ) = max u(x) ≥ 0 =⇒ (T u)(x0 ) ≤ 0. (PMB) x∈D

Proof. Indeed, we have, by formula (11.4) and condition (11.5),         (T u)(x0 ) = η(x0 )u(x0 ) + ζ(x0 ) · u(x0 ) + t(x0 , dy) u(y) D

  N −1   ∂u   τβ (x0 , y) u(x0 ) + (x ) − χβj (y) − χβj (x0 ) 0 ∂χβj j=1 β=1    m  = η(x0 )u(x0 ) + t(x0 , dy) u(y) − τβ (x0 , y) · u(x0 ) m 

D

β=1

11.2 Positive Boundary Maximum Principles

329

    m     = η(x0 ) + t(x0 , dy) 1 − τβ (x0 , y) · u(x0 )  + D

D

β=1

t(x0 , dy) (u(y)

−

u(x0 ))

≤ 0. This proves the positive boundary maximum principle (PMB) for the boundary operator T . The proof of Theorem 11.3 is complete.   11.2.2 Boundary Maximum Principles for Ventcel’ operators Secondly, we can prove the following theorem for Ventcel’ boundary operators Γ: Theorem 11.4. For a linear boundary operator Γ : C 2 (D) −→ B(∂D), the following two assertions (I) and (II) are equivalent: (I) The boundary operator Γ satisfies the positive boundary maximum principle: x0 ∈ ∂D, u ∈ C 2 (D) and u(x0 ) = max u(x) ≥ 0 =⇒ (Γ u)(x0 ) ≤ 0. (PMB) x∈D

(II) The boundary operator Γ is of the form (Γ u)(x ) := (Λu)(x ) + (T u)(x )    ∂u   (x ) + Qu(x ) + (T u)(x ) = μ(x ) ∂n

(11.6)

N −1 N −1   ∂u  ∂2u ∂u  ij   (x ) + α (x ) (x ) + β i (x ) (x ) = μ(x ) ∂n ∂χ ∂χ ∂χ i j i i,j=1 i=1 









+ γ(x )u(x ) + η(x )u(x ) +

N −1  j=1

∂u  ζ (x ) (x ) + ∂χj j





 t(x , dy) u(y) 

D

  m N −1   ∂u   β β     − χj (y) − χj (x ) τβ (x , y) u(x ) + (x ) ∂χβj j=1 β=1 for every x ∈ U ∩ ∂D. Here:

330

11 Boundary Operators and Boundary Maximum Principles

(a) The operator Q is a second-order, degenerate elliptic differential operator on the boundary ∂D with non-positive principal symbol. In other words, ij the 2 α are the components of a symmetric contravariant tensor of type 0 on ∂D satisfying the condition N −1 

αij (x )ξi ξj ≥ 0

for all x ∈ ∂D and ξ  =

&N −1 j=1

ξj dxj ∈ Tx∗ (∂D).

i,j=1

(b) Q1(x ) = γ(x ) ∈ B(∂D) and γ(x ) ≤ 0 on ∂D. (c) μ(x ) ∈ B(∂D) and μ(x ) ≥ 0 on ∂D. (d) n = (n1 , n2 , . . . , nN ) is the unit inward normal to ∂D. (e) The operator T is a Ventcel’–L´evy boundary operator, that is, (e.1) t(x , dy) is a Ventcel’ kernel on the manifold D. (e.2) ζ(x ) is a bounded, Borel measurable vector field on ∂D; more precisely, we have, in a local chart (U, χ), ζ(x ) · u =

N −1 

ζ j (x )

j=1

∂u ∂χj

with ζ j ∈ Bloc (U ∩ ∂D).

(e.3) η(x ) is a bounded, Borel measurable function on ∂D which satisfies condition (11.3). Definition 11.5. The boundary operator Λ = μ(x )

∂ + Q : C 2 (D) −→ B(∂D) ∂n

is called a Ventcel’–Viˇsik boundary operator (see [15, p. 436, formule (II.2.1)], [138]). The boundary operator Γ = Λ + T = μ(x )

∂ + Q + T : C 2 (D) −→ B(∂D) ∂n

is called a Ventcel’ boundary operator (see [15, p. 442, formule (II.2.16)], [138]). The proof of Theorem 11.4 is given in the next Subsubsections 11.2.2 and 11.2.2, due to its length. Proof of Theorem 11.4, Part (i) • Assertion (II) =⇒ Assertion (I): Assume that x0 ∈ ∂D, u ∈ C 2 (D) and u(x0 ) = max u(x) ≥ 0. x∈D

11.2 Positive Boundary Maximum Principles

331

Then we remark that

∂u  (x ) ≤ 0, ∂n 0 since n is the unit inward normal to ∂D. Therefore, we have, by formula (11.6) and condition (11.5), (Γ u)(x0 ) = (Λu)(x0 ) + (T u)(x0 )    ∂u   = μ(x0 ) (x ) + Qu(x0 ) + (T u)(x0 ) ∂n 0 = μ(x0 ) +

N −1 N −1   ∂u  ∂2u ∂u  (x0 ) + αij (x0 ) (x0 ) + β i (x0 ) (x0 ) ∂n ∂χ ∂χ ∂χ i j i i,j=1 i=1

γ(x0 )u(x0 )

+

η(x0 )u(x0 )

+

N −1 

ζ

j

(x0 )

j=1

−

m 

∂u  (x ) + ∂χj 0



D



t(x0 , dy)

u(y)

  N −1   ∂u  β β    χj (y) − χj (x0 ) u(x0 ) + (x0 ) ∂χβj j=1

τβ (x0 , y)

β=1

N −1  ∂u  ∂2u (x ) + αij (x0 ) (x0 ) + γ(x0 )u(x0 ) ∂n ∂χ ∂χ i j i,j=1    m       t(x0 , dy) u(y) − τβ (x0 , y) · u(x0 ) + η(x0 )u(x0 ) +

= μ(x0 )

D

β=1

    m  ≤ η(x0 ) + t(x0 , dy) 1 − τβ (x0 , y) · u(x0 )  + D

D

β=1

t(x0 , dy) (u(y) − u(x0 ))

≤ 0. This proves the positive boundary maximum principle (PMB) for the boundary operator Γ .   Proof of Theorem 11.4, Part (ii) • Assertion (I) =⇒ Assertion (II): The proof is divided into five steps. Step 1: First, by the positive boundary maximum principle (PMB) for Γ = Λ + T we obtain that u ∈ C0∞ (D \ {x }), u ≥ 0 on D =⇒ (Γ u)(x ) ≥ 0. Hence there exists a positive Borel kernel t(x , dy) of ∂D into D which satisfies the condition (NV.1) and the formula

332

11 Boundary Operators and Boundary Maximum Principles 



(Γ u)(x ) =

t(x , dy) u(y) whenever u ∈ C 2 (D) and x ∈ supp u. (11.7)

D

Step 2: Moreover, the kernel t(x , dy) satisfies the condition (NV.3). Indeed, let f be an arbitrary non-negative function on D. Then, for each point x ∈ ∂D there exist a neighborhood V of x and a non-negative function u ∈ C 2 (D) vanishing on V such that (1 − σ(x , y)) f (y) ≤ u(y) for all y ∈ M and x ∈ V . Hence we have, by formula (11.7),   t(x , dy) (1 − σ(x , y)) f (y) ≤ t(x , dy) u(y) = (Γ u)(x ) for all x ∈ V . D

D

This proves the desired condition (NV.3), since Γ u ∈ B(∂D). Step 3: Now we verify the condition (NV.2) for the kernel t(x , dy). Let (U, χ) be an arbitrary local chart on D (see Figure 11.2 below) such that U ∩ ∂D = ∅, χ(U ) is bounded in RN .

xN

∂D

D

Ω = χ(U )

U n

χ x

Fig. 11.2. The local chart (U, χ) on D

Substep 3.1: First, we show the following two assertions (i) and (ii): (i) For every non-negative function f ∈ C(D) with supp f ⊂ U , we have the assertion 

 2  N −1  t(x , dy) f (y) χN (y)+ χj (y)−χj (x ) ∈ Bloc (U ∩∂D). (11.8)

U

j=1

(ii) There exist functions bij (x ), bi (x ) and b(x ) ∈ Bloc (U ∩ ∂D) such that we have, for every function u ∈ C 2 (D) with supp u ⊂ U , (Γ u)(x )

(11.9)

11.2 Positive Boundary Maximum Principles

= μ(x )

333

N −1 N −1   ∂u  ∂2u ∂u  (x ) + bij (x ) (x ) + bi (x ) (x ) ∂χN ∂χi ∂χj ∂χi i,j=1 i=1

+ b(x )u(x )    N −1   ∂u   t(x , dy) u(y) − u(x ) − (x ) χj (y) − χj (x ) . + ∂χj U j=1 Here: N −1 

bij (x )ξi ξj ≥ 0

for all ξ  = (ξ1 , . . . , ξN −1 ) ∈ RN −1 ,

(11.10)

i,j=1

and

μ(x ) ≥ 0.

(11.11)

Now we let Ω = χ(U ), 0 of RN (see Figure 11.3 below) such that and choose an open subset Ω 0 ∩ RN . χ(U ) = Ω +

xN Ω = Ω ∩ RN +

x Ω

Fig. 11.3. The set Ω = χ(U ) and the open set Ω

Then we define a linear operator (see Figure 11.4 below) 0 −→ B(Ω) 0 Γ0 : C02 (Ω) by the formula

334

11 Boundary Operators and Boundary Maximum Principles Γ

C02 (Ω) −−−−→ ⏐ ⏐ χ∗

B(Ω) ⏐ −1 ⏐(χ )∗

2 C(0) (U ) −−−−→ Bloc (U ∩ ∂D) Γ

Fig. 11.4. The operators Γ and Γ

 Γ0ϕ(z) =

  Γ (ϕ 0 ◦ χ) χ−1 (z) for z ∈ ∂Ω, 0 \ ∂Ω. 0 for z ∈ Ω

Here ϕ 0 is the restriction of ϕ to the set Ω = χ(U ). 2 (U ) → Bloc (U ∩ ∂D) satisfies the condition (PMB), it is easy Since Γ : C(0) 0 to see that Γ satisfies the positive maximum principle ◦

0 ϕ ∈ C 2 (Ω) 0 and ϕ(z) = max ϕ(w) ≥ 0 =⇒ Γ0ϕ(z) ≤ 0. z ∈ Ω, 0  w∈Ω

(PM)

Therefore, by applying Theorem 4.6 to our situation we obtain that the func0 and z ∈ Ω 0 is of the form tion Γ0ϕ(z) for ϕ ∈ C02 (Ω) Γ0ϕ(z) =

N 

2  0bij (z) ∂ ϕ (z) + 0bi (z) ∂ϕ (z) + 0b(z)ϕ(z) ∂zi ∂zj ∂zi i,j=1 i=1 N

(11.12)

   N  ∂ϕ 0 t(z, dw) ϕ(w) − σ(z, w) ϕ(z) + (z)(wj − zj ) . ∂zj  Ω j=1



+ Here:

0 such that (a) 0bij (z), 0bi (z) and 0b(z) ∈ Bloc (Ω) N 

0bij (z) ≥ 0 for all (z, ξ) ∈ Ω 0 × RN .

(11.13)

i,j=1

0 (b) σ(z, w) is a local unity function on Ω. 0 0 (c) t(z, dw) is a L´evy kernel on Ω such that 0b(z) +

  Ω

0 t(z, dw) (1 − σ(z, w)) ≤ 0

◦

0 for all z ∈ Ω.

(11.14)

We remark that the kernels t(x , dy) and 0 t(z, dw) are related as follows:   0 for all z ∈ ∂Ω and Z ∈ BΩ . t(z, Z) = t χ−1 (z), χ−1 (Z)

11.2 Positive Boundary Maximum Principles

335

Indeed, it suffices to note the following formulas with w = χ(y) and x = χ−1 (z) ∈ ∂D:    0 0 t(z, dw) ϕ(w) = Γ ϕ(z) = Γ (ϕ 0 ◦ χ)(x ) = t(x , dy) (ϕ 0 ◦ χ)(y)  D Ω  t(x , dy) (ϕ 0 ◦ χ)(y) = U

0 with z ∈ supp ϕ. for all ϕ ∈ C02 (Ω) In particular, it should be emphasized that the measure 0 t(z, ·) for z ∈ ∂Ω is supported on the Ω = χ(U ). 0 satisfies condition (NS2), we Moreover, since the L´evy kernel 0 t(z, ·) on Ω have the assertion  2 t(x , dy) (χj (y) − χj (x )) (11.15) U  2 0 = t(z, dw) (wj − zj ) ∈ Bloc (U ∩ ∂D) for all 1 ≤ j ≤ N − 1. Ω

Substep 3.2: We show that  t(x , dy) f (y) χN (y) ∈ B(U ∩ ∂D)

(11.16)

U

for every non-negative function f ∈ C(D) with supp f ⊂ U . First, we prove that 0bN (z) ≥ 0 on ∂Ω.

(11.17)

To do so, for z ∈ ∂Ω we let ϕ1 (w) = σ(z, w) wN

0 for w ∈ Ω.

Then it is easy to see the following formulas: • ϕ1 (z) = σ(z, z) 0 = 0, ∂ϕ1 (z) = 0 for all 1 ≤ i ≤ N − 1, • ∂wi ∂ϕ1 (z) = 1, • ∂wN and w − z = (w1 − z1 , . . . , wN −1 − zN −1 , wN ) . Hence, by applying the positive boundary maximum principle (PMB) to the function 0 −ϕ1 (w) = −σ(z, w) wN for w ∈ Ω, we obtain that

336

11 Boundary Operators and Boundary Maximum Principles

0 ≤ Γ0ϕ1 (z) = 0bN (z)    N    ∂ϕ1 0 t(z, dw) ϕ1 (w) − σ(z, w) ϕ1 (z) + (z)(wj − zj ) + ∂zj Ω j=1    N 0 0 = b (z) + t(z, dw) σ(z, w) wN − σ(z, w) wN Ω

= 0bN (z) on ∂Ω, so that,

0bN (z) ≥ 0

on ∂Ω.

Secondly, for z ∈ ∂Ω we let ϕ2 (w) = σ1 (z, w) wN ,

0 w ∈ Ω,

0 that will be chosen later on. where σ1 (z, w) is a local unity function on Ω Then it is easy to see the following formulas: • ϕ2 (z) = σ(z, z) 0 = 0, ∂ϕ2 (z) = 0 for all 1 ≤ i ≤ N − 1, • ∂wi ∂ϕ2 (z) = 1. • ∂wN Hence we have the formula Γ0ϕ2 (z)

(11.18)    N  ∂ϕ2 N 0 0 t(z, dw) ϕ2 (w) − σ(z, w) ϕ1 (z) + (z)(wj − zj ) = b (z) + ∂zj Ω j=1    N 0 0 = b (z) + t(z, dw) σ1 (z, w) wN − σ(z, w) wN on ∂Ω. 

Ω

However, by virtue of inequality (11.17) it follows that  0 Γ0ϕ2 (z) ≥ t(z, dw) (σ1 (z, w) − σ(z, w)) wN

on ∂Ω.

Ω

Therefore, by shrinking the support supp σ of σ to the diagonal   0 , ΔΩ = (z, z) : z ∈ Ω we obtain from an application of the Lebesgue dominated convergence theorem that

11.2 Positive Boundary Maximum Principles

Γ0ϕ2 (z) ≥



0 t(z, dw) σ1 (z, w) wN

on ∂Ω.

337

(11.19)

Ω

Indeed, it suffices to note that σ(z, w) wN −→ 0

as w → z.

Hence, by assertion (11.19) we find that  t(x , dy) f (y) χN (y) U  0 = t(z, dw) f (χ−1 (w)) wN χ(U)  0 = t(z, dw) σ1 (z, w) wN f (χ−1 (w)) ∈ B(U ∩ ∂D) Ω

for every non-negative function f ∈ C(D) with supp f ⊂ U , if we choose the local unity function σ1 (z, w) such that σ1 (z, w) = 1

on supp (f ◦ χ−1 ).

The desired condition (NV.2) for the kernel t(x , dy) follows by combining assertions (11.15) and (11.16). That is, for every local chart (U, χ) of D such that U ∩ ∂D = ∅ and for every non-negative function f ∈ C(D) with supp f ⊂ U , the function U ∩ ∂D  x −→



 2  N −1  t(x , dy) f (y) χN (y) + χj (y) − χj (x )

U

j=1

belongs to the space Bloc (U ∩ ∂D). Summing up, we have proved that t(x , dy) is a L´evy kernel on the manifold D. Step 4: Now we prove assertion (ii). By condition (NV.2), we can rewrite formula (11.12) in the form    ∂ϕ N 0 0 0 Γ ϕ(z) = b (z) − (z) (11.20) t(z, dw) wN ∂z N Ω +

N −1 2  0bij (z) ∂ ϕ (z) + 0bi (z) ∂ϕ (z) + 0b(z)ϕ(z) ∂z ∂z ∂zi i j i,j=1 i=1 N 

   N −1  ∂ϕ 0 + (z)(wj − zj ) . t(z, dw) ϕ(w) − σ(z, w) ϕ(z) + ∂zj Ω j=1 

Substep 4-1: We show assertion (11.11), that is,  0 μ 0(z) = 0bN (z) − t(z, dw) σ(z, w) wN ≥ 0 on ∂Ω. Ω

(11.21)

338

11 Boundary Operators and Boundary Maximum Principles

Indeed, by applying the positive boundary maximum principle (PMB) to the function −ϕ2 (w) we obtain that 0 ≤ Γ0ϕ2 (z) = 0bN (z)     N  ∂ϕ2 0 (z)(wj − zj ) + t(z, dw) ϕ2 (w) − σ(z, w) ϕ2 (z) + ∂zj Ω j=1    0 = 0bN (z) + t(z, dw) σ1 (z, w) wN − σ(z, w) wN  Ω 0 =μ 0(z) + t(z, dw) σ1 (z, w) wN on ∂Ω, Ω



so that

0 t(z, dw) σ1 (z, w) wN

μ 0(z) ≥ −

on ∂Ω.

Ω

Therefore, by shrinking the support supp σ1 of σ1 to the diagonal ΔΩ we find from the condition (NV.2) that μ 0(z) ≥ 0

on ∂Ω.

Substep 4-2: We show assertion (11.10). First, we prove that 0bN N (z) ≡ 0 on ∂Ω.

(11.22)

In view of condition (11.13), we assume, to the contrary, that 0bN N (z0 ) > 0 for some point z0 ∈ ∂Ω. For z ∈ ∂Ω, we let ϕ3 (w) = σ1 (z, w) θ(wN )

0 for every w ∈ Ω,

0 and θ(wN ) is a non-negative where σ1 (z, w) is a local unity function on Ω 2 function in C [0, ∞) that will be chosen later on. By applying the positive boundary maximum principle (PMB) to the function −ϕ3 (w), we obtain from formula (11.20) that Γ0ϕ3 (z0 ) = μ 0(z0 ) θ (0) + 0b(z0 ) θ(0) + 0bN N (z0 ) θ (0)  0 + t(z0 , dw) σ1 (z0 , w) (θ(wN ) − θ(0)) Ω

≥ 0. However, we can choose a function θ(wN ) such that θ(0) = sup θ > 0, [0,∞)]

(11.23)

11.2 Positive Boundary Maximum Principles

339

θ (0) = 0, θ (0) < 0, and further that

0b(z0 ) θ(0) + 0bN N (z0 ) θ (0) < 0,

(11.24)



if θ(0) is sufficiently small and |θ (0)| is sufficiently large. Therefore, by shrinking the support supp σ1 of σ1 to the diagonal ΔΩ we obtain from inequality (11.24) that Γ0ϕ3 (z0 ) = 0b(z0 ) θ(0) + 0bN N (z0 ) θ (0) +



0 t(z0 , dw) σ1 (z0 , w) (θ(wN ) − θ(0))

Ω

< 0. This contradicts inequality (11.23). Secondly, we show that 0bN 1 (z) = . . . = 0bN N −1 (z) ≡ 0

on ∂Ω.

(11.25)

We assume, to the contrary, that 0bN j (z0 ) = 0

for some z0 = (z1 , . . . , zN −1 , 0) ∈ ∂Ω and 1 ≤ j ≤ N − 1.

For z ∈ ∂Ω, we let ϕ4 (w) = σ1 (z, w) θ(wj ) wN

0 for every w ∈ Ω,

0 and θ(wj ) is a non-negative where σ1 (z, w) is a local unity function on Ω function in C 2 [0, ∞) that will be chosen later on. By applying the positive boundary maximum principle (PMB) to the function −ϕ4 (w), we obtain from formula (11.20) that Γ0ϕ4 (z0 ) =μ 0(z0 ) θ(zj ) + 0bN j (z0 ) θ (zj ) +



(11.26) 0 t(z0 , dw) σ1 (z, w) θ(wj ) wN

Ω

≥ 0. However, we can choose a function θ(wj ) such that θ (zj ) < 0 θ (zj ) > 0

and further that

if 0bN j (z) > 0, if 0bN j (z) < 0,

μ 0(z0 ) θ(zj ) + 0bN j (z0 ) θ (zj ) < 0,

(11.27)

340

11 Boundary Operators and Boundary Maximum Principles

if θ(zj ) is sufficiently small and |θ (zj )| is sufficiently large. Therefore, by shrinking the support supp σ1 of σ1 to the diagonal ΔΩ we obtain from inequality (11.27) that Γ0ϕ4 (z0 )



=μ 0(z0 ) θ(zj ) + 0bN j (z0 ) θ (zj ) +

0 t(z0 , dw) σ1 (z0 , w) θ(wj ) wN

Ω

< 0. This contradicts inequality (11.26). Therefore, the desired assertion (11.10) follows by combining assertions (11.13), (11.22) and (11.25). Step 5: Finally, we prove assertion (11.9). Now we consider the following: (1) Let t(x , dy) be the Ventcel’ kernel on the manifold D constructed in Step 3. (2) Let ζ(x ) be a bounded, Borel measurable vector field on the boundary ∂D. More precisely, we have, in a local chart (Uβ , χβ ), ζ(x ) · u =

N −1  j=1

ζ j (x )

∂u ∂χβj

with ζ j ∈ Bloc (Uβ ∩ ∂D).

(3) Let η(x ) be a bounded, Borel measurable function on the boundary ∂D such that    m  t(x , dy) 1 − τβ (x , y) ≤ 0 on ∂D. η(x ) + D

β=1

Then we can define a Ventcel’–L´evy boundary operator T0 : C 2 (D) −→ B(∂D) by formula (11.4) T0u(x ) = η(x )u(x ) + ζ(x ) · u(x ) +



 t(x , dy) u(y) 

D

  N −1  ∂u  β β    τβ (x , y) u(x ) + (χj (y) − χj (x )) β (x ) . − ∂χj j=1 β=1 m 

We remark that the above condition (3) is equivalent to the following: (T01)(x ) ≤ 0 If we let

on ∂D.

11.3 Notes and Comments

341

⎧   ⎨T u := T0u + Γ 1 − T01 u,   ⎩Λu := Γ u − T0u − Γ 1 − T01 u, then we can obtain the desired decomposition (11.9) of the boundary operator Γ = T + Λ. Now the proof of Theorem 11.4 is complete.  

11.3 Notes and Comments Section 11.1: Theorem 11.3 is taken from Bony–Courr`ege–Priouret [15, p. 441, assertion (PMB)]. Section 11.2: Theorem 11.4 is taken from Bony–Courr`ege–Priouret [15, p. 441, Th´eor`eme X]. This chapter is based on the lecture entitled A class of hypoelliptic Vishik– Wentzell boundary vale problems delivered at Functional Analysis Methods for Partial Differential Equations, Centro Polifunzionale, Cesena, Italy, on June 27, 2019.

Part V

Feller Semigroups for Elliptic Waldenfels Operators

12 Proof of Theorem 1.5 - Part (i) -

Part V (Chapters 12 through 14) is devoted to the proof of generation theorems of Feller semigroup for second-order, elliptic Waldenfels integrodifferential operators. This Chapter 12 and the next Chapter 13 are devoted to the proof of Theorem 1.5 and Theorem 1.6 (cf. [121], [123]). In this chapter we prove part (i) of Theorem 1.5. In the proof we make use of Sobolev’s imbedding theorems (Theorems 4.8 and 4.9) and a λ-dependent localization argument due to Masuda [78] (Lemma 12.2) in order to adjust the resolvent estimate  c (ε)   p −1  (Ap − λI)  ≤ |λ|

for all λ ∈ Σp (ε)

(1.9)

to obtain the desired estimate   (A − λI)−1  ≤ c(ε) |λ|

for all λ ∈ Σ(ε).

(1.11)

Here recall that the operator Ap : Lp (D) −→ Lp (D) is a densely defined, closed linear operator defined by the following: (a) The domain of definition D (Ap ) is the set   D (Ap ) = u ∈ H 2,p (D) = W 2,p (D) : Lu = 0 on ∂D .

(1.8)

(b) Ap u = Au for every u ∈ D (Ap ). Furthermore, the operator     A : C0 D \ M −→ C0 D \ M © Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 12

346

12 Proof of Theorem 1.5 - Part (i) -

is a densely defined, closed linear operator defined by the following: (c) The domain of definition D (A) is the set       D (A) = u ∈ C0 D \ M : Au ∈ C0 D \ M , Lu = 0 on ∂D . (1.10) (d) Au = Au for every u ∈ D (A). The proof of part (i) of Theorem 1.5 can be flowcharted as in Table 12.1 below. Theorem 4.9 (Gagliardo–Nirenberg) Lemma 12.2 (inequality (12.8)) Lemma 12.1 (inequality (12.3)) Part (i) of Theorem 1.5 Theorem 4.8 (Sobolev) Lemma 12.5 (unique solvability in C0 (D \ M )) Theorem 1.4 (unique solvability in Lp (D))

Table 12.1. A flowchart for the proof of part (i) of Theorem 1.5

  12.1 Space C0 D \ M First, we consider a one-point compactification K∂ = K ∪ {∂} of the space K = D \ M. We say that two points x and y of D are equivalent modulo M if x = y or x, y ∈ M ; we then write x ∼ y. It is easy to verify that this relation ∼ enjoys the so-called equivalence laws. We denote by D/M the totality of equivalence classes modulo M . On the set D/M we define the quotient topology induced by the projection q : D −→ D/M. Namely, a subset O of D/M is defined to be open if and only if the inverse image q −1 (O) of O is open in D. It is easy to see that the topological space D/M is a one-point compactification of the space D \ M and that the point at infinity ∂ corresponds to the set M (see Figure 12.1 below):  K∂ := D/M, ∂ := M. Furthermore, we obtain the following two assertions (i) and (ii):

12.2 Proof of Part (i) of Theorem 1.5

347

∂D

D

q

D/M

M

∂

Fig. 12.1. The compactification D/M of D \ M

(i) If u ˜ is a continuous function defined on K∂ , then the function u ˜ ◦ q is continuous on D and constant on M . (ii) Conversely, if u is a continuous function defined on D and constant on M , then it defines a continuous function u ˜ on K∂ . In other words, we have the following isomorphism:   C(K∂ ) ∼ = u ∈ C(D) : u(x) is constant on M .

(12.1)

Now we introduce a closed subspace of C(K∂ ) as in Subsection 4.1.1: C0 (K) = {u ∈ C(K∂ ) : u(∂) = 0} . Then we have, by assertion (12.1),     C0 (K) ∼ = C0 D \ M = u ∈ C(D) : u(x) = 0 on M .

(12.2)

12.2 Proof of Part (i) of Theorem 1.5 The proof is carried out in a chain of auxiliary lemmas. Step (I): We begin with the following fundamental inequality (see [121, p. 115, Lemma 8.3]): Lemma 12.1. Let N < p < ∞. Assume that the following two conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D. Then, for every ε > 0, there exists a constant rp (ε) > 0 such that if λ = r2 ei θ with r ≥ rp (ε) and −π + ε ≤ θ ≤ π − ε, we have, for all u ∈ D (Ap ), 1/2

|λ|

|u|C 1 (D) + |λ| · |u|C(D) ≤ cp (ε) |λ|

N/2p

(A − λ)u p ,

with a constant cp (ε) > 0. Here   2,p  ∂u  D (Ap ) = u ∈ H (D) : Lu = μ(x ) + γ(x )u = 0 . ∂n

(12.3)

348

12 Proof of Theorem 1.5 - Part (i) -

Proof. First, we know from [121, p. 102, estimate (7.1)] that the estimate |u|2,p + |λ|

1/2

|u|1,p + |λ| u p ≤ cp (ε) (Ap − λI)u p

holds true for all u ∈ D (Ap ), where

u p = u Lp (D) ,

(12.4)

  |u|2,p = ∇2 uLp (D) .

|u|1,p = ∇u Lp (D) ,

On the other hand, by applying Theorem 4.9 with p := r > N , θ := N/p and ν := 0 we obtain from the Gagliardo–Nirenberg inequality (4.6) that N/p

1−N/p

|u|C(D) ≤ C |u|1,p u p

for all u ∈ H 2,p (D).

(12.5)

Here and in the following the letter C denotes a generic positive constant depending on p and ε, but independent of u and λ. By using estimate (12.4), we obtain from inequality (12.5) that N/p

1−N/p

|u|C(D) ≤ C |u|1,p u p N/p  1−N/p  |λ|−1 (A − λ)u p ≤ C |λ|−1/2 (A − λ)u p −1+N/2p

= C |λ|

(A − λ)u p .

This proves that |λ| · |u|C(D) ≤ C |λ|

N/2p

(A − λ)u p

for all u ∈ D (Ap ).

(12.6)

Similarly, by applying inequality (12.5) to the functions Di u ∈ H 1,p (D)

for 1 ≤ i ≤ n,

we obtain that 1−N/p 1−N/p |Di u|C(D) ≤ C |Di u|N/p ≤ C |u|N/p 1,p Di u p 2,p |u|1,p N/p  1−N/p  −1/2 |λ| ≤ C (A − λ)u p

(A − λ)u p

= C |λ|−1/2+N/2p (A − λ)u p . This proves that |λ|1/2 |u|C 1 (D) ≤ C |λ|N/2p (A − λ)u p

for all u ∈ D (Ap ).

(12.7)

Therefore, the desired inequality (12.3) follows by combining inequalities (12.6) and (12.7). The proof of Lemma 12.1 is complete.   The next lemma proves the resolvent estimate (1.7):

12.2 Proof of Part (i) of Theorem 1.5

349

Lemma 12.2. Assume that conditions (A) and (B) are satisfied. Then, for every ε > 0, there exists a constant r(ε) > 0 such that if λ = r2 ei θ with r ≥ r(ε) and −π + ε ≤ θ ≤ π − ε, we have, for all u ∈ D (A), |λ|1/2 |u|C 1 (D) + |λ| · |u|C(D) ≤ c(ε)|(A − λ)u|C(D) ,

(12.8)

with a constant c(ε) > 0. Here       D (A) = u ∈ C0 D \ M : Au ∈ C0 D \ M , Lu = 0 on ∂D . Proof. We shall make use of a λ-dependent localization argument due to Masuda [78] in order to adjust the term (A−λ)u p in inequality (12.3) to obtain inequality (12.8). First, we remark that A ⊂ Ap

for all 1 < p < ∞.

Indeed, since we have, for any u ∈ D (A), u ∈ C(D) ⊂ Lp (D), Au ∈ C(D) ⊂ Lp (D) and Lu = 0 on ∂D, it follows from an application of Theorem 6.21 and [123, Theorem 7.1] (see also [68, Theorem 3.1]) that u ∈ H 2,p (D). (1) Let x0 be an arbitrary point of the closure D = D ∪ ∂D. If x0 is a boundary point and if χ is a smooth coordinate transformation such that χ maps B (x0 , η0 ) ∩ D into B(0, δ) ∩ RN + and flattens a part of the boundary ∂D into the plane xN = 0 (see Figure 12.2 below), then we let • G0 = B (x0 , η0 ) ∩ D,

• G = B (x0 , η) ∩ D for 0 < η < η0 , • G = B (x0 , η/2) ∩ D for 0 < η < η0 , where B(x, η) denotes the open ball of radius η about x (see Figure 12.3 below). Similarly, if x0 is an interior point and if χ is a smooth coordinate transformation such that χ maps B (x0 , η0 ) into B(0, δ), then we let (see Figure 12.4 below) • G0 = B (x0 , η0 ) , • G = B (x0 , η) for 0 < η < η0 , • G = B (x0 , η/2) for 0 < η < η0 .

350

12 Proof of Theorem 1.5 - Part (i) xN

∂D

D

B(x0 , η0 )

B(0, δ)

n

χ x0

0

x = (x1 , . . . , xN −1 )

Fig. 12.2. The coordinate transformation χ maps B (x0 , η0 ) into B(0, δ)

D

G0 = B(x0 , η0 )

∂D

.

G G

• x0

Fig. 12.3. The half-balls G0 , G and G G0 = B(x0 , η0 ) ∂D D G G

x0

Fig. 12.4. The open balls G0 , G and G

(2) Now we take a function θ(t) in C0∞ (R) such that θ(t) equals one near the origin, and define a function ϕ(x) = θ(|x |2 ) θ(xN ) for x = (x , xN ).

12.2 Proof of Part (i) of Theorem 1.5

351

Here we may assume that the function ϕ(x) is chosen so that  supp ϕ ⊂ B(0, 1), ϕ(x) = 1 on B(0, 1/2). We introduce a localizing function        x − x0 xN − t |x − x0 |2 ϕ0 (x, η) ≡ ϕ θ =θ η η2 η for x0 = (x0 , t) ∈ D. 

We remark that

supp ϕ0 ⊂ B (x0 , η) , ϕ0 (x, η) = 1 on B (x0 , η/2).

Then, for the localizing function ϕ0 (x, η) we have the following claim: Claim 12.3. If u ∈ D (A), then it follows that ϕ0 (x, η)u ∈ D (Ap ) for all 1 < p < ∞. Proof. (i) First, we recall that u ∈ H 2,p (D) for all 1 < p < ∞. Hence we have the assertion ϕ0 (x, η)u ∈ H 2,p (D). (ii) Secondly, it is easy to verify (see Figure 12.5 below) that the function ϕ0 (x, η)u, x ∈ D, satisfies the boundary condition L(ϕ0 (x , η)u) = 0

on ∂D.

Indeed, this is obvious if we have the condition supp (ϕ0 (x, η)u) ⊂ B (x0 , η)

for x0 ∈ D.

Moreover, if we have the condition supp (ϕ0 (x, η)u) ⊂ B (x0 , η) ∩ D then it follows that

for x0 ∈ ∂D,

     |x − x0 |2 1  ∂  (ϕ0 (x, η)) = θ (0) · θ = 0, ∂xN η η2 xN =0

since θ (0) = 0. This proves that

352

12 Proof of Theorem 1.5 - Part (i) -

D

∂D

B(x0 , η)

B(x0 , η)

x0 x0

Fig. 12.5. The open balls B (x0 , η) in D and on ∂D

∂ (ϕ0 (x , η)) = 0 ∂n

on ∂D.

Therefore, we have the assertion ∂ (ϕ0 (x , η)u) + γ(x )ϕ0 (x , η)u ∂n   ∂    = ϕ0 (x , η)(Lu) + μ(x ) (ϕ0 (x , η)) u ∂n

L(ϕ0 (x, η)u) = μ(x )

= 0 on ∂D, since Lu = 0 on ∂D. Summing up, we have proved that ϕ0 (x, η)u ∈ D (Ap )

for all 1 < p < ∞.

The proof of Claim 12.3 is complete.   (3) Now we take a positive number p such that N < p < ∞. By virtue of Claim 12.3, we can apply the fundamental inequality (12.3) to ϕ0 (x, η) u with u ∈ D (A) to obtain that |λ|1/2 |u|C 1 (G ) + |λ| · |u|C(G )

(12.9)

≤ |λ|1/2 |ϕ0 (x, η)u|C 1 (G ) + |λ| · |ϕ0 (x, η)u|C(G ) = |λ|1/2 |ϕ0 (x, η)u|C 1 (D) + |λ| · |ϕ0 (x, η)u|C(D) ≤ C|λ|N/2p (A − λ)(ϕ0 (x, η)u) Lp (D) = C|λ|N/2p (A − λ)(ϕ0 (x, η)u) Lp (G ) since we have the assertions

for 0 < η < η0 ,

12.2 Proof of Part (i) of Theorem 1.5



353

ϕ0 (x, η) = 1 on G , supp (ϕ0 (x, η)u) ⊂ G .

However, we have the formula (A − λ) (ϕ0 (x, η)u) = ϕ0 (x, η) ((A − λ)u) + [A, ϕ0 (x, η)] u,

(12.10)

where [A, ϕ0 (x, η)] is the commutator of A and ϕ0 (x, η) defined by the formula [A, ϕ0 (x, η)] u (12.11) = A (ϕ0 (x, η)u) − ϕ0 (x, η)Au ⎞ ⎛ N N N 2    ∂ϕ0 ∂u ∂ ϕ0 ∂ϕ0 ⎠ =2 u. aij (x) +⎝ aij (x) + bi (x) ∂x ∂x ∂x ∂x ∂xi i j i j i,j=1 i,j=1 i=1 Here we need the following elementary inequality: Claim 12.4. We have, for all v ∈ C j (G ), j = 0, 1, 2,

v H j,p (G ) ≤ |G |

1/p

v C j (G ) ,

where |G | denotes the measure of G . Proof. It suffices to note that we have, for all w ∈ C(G ),  p |w(x)|p dx ≤ |G | |w|C(G ) . G

This proves Claim 12.4.   Since we have (see Figure 12.3), for some positive constant c, |G | ≤ |B (x0 , η)| ≤ cη N , it follows from an application of Claim 12.4 that

ϕ0 (x, η) ((A − λ) u) Lp (G ) ≤ c1/p η N/p |(A − λ) u|C(G )

(12.12)

for 0 < η < η0 . Furthermore, we remark that   |Dα ϕ0 (x, η)| = O η −|α|

as η ↓ 0.

Hence it follows from an application of Claim 12.4 that we have, for 0 < η < η0 ,    ∂ϕ0 ∂u  C −1+N/p  •  |u|C 1 (G ) , (12.13)  ∂xi ∂xj  p  ≤ η |u|1,p,G ≤ Cη L (G )

354

12 Proof of Theorem 1.5 - Part (i) -

 2   ∂ ϕ0  C −2+N/p  •  u |u|C(G ) ,  ∂xi ∂xj  p  ≤ η 2 |u|Lp (G ) ≤ Cη L (G )    ∂ϕ0  C  •  u ≤ |u|Lp (G ) ≤ Cη −1+N/p |u|C(G ) . ∂xi Lp (G ) η

(12.14) (12.15)

By using inequalities (12.13), (12.14) and (12.15), we obtain from formula (12.11) that

[A, ϕ0 (x, η)] u Lp (G ) (12.16)   ≤ C η −1+N/p |u|C 1 (G ) + η −2+N/p |u|C(G ) + η −1+N/p |u|C(G )   for 0 < η < η0 . ≤ C η −1+N/p |u|C 1 (D) + η −2+N/p |u|C(D) In view of formula (12.10), it follows from inequalities (12.12) and (12.16) that

(A − λ)(ϕ0 (x, η)u) Lp (G )

(12.17)

≤ ϕ0 (x, η)((A − λ)u) Lp (G ) + [A, ϕ0 (x, η)]u Lp (G )   ≤ Cη N/p |(A − λ)u|C(G ) + η −1 |u|C 1 (D) + η −2 |u|C(D) for 0 < η < η0 . Therefore, by combining inequalities (12.9) and (12.17) we obtain that |λ|1/2 |u|C 1 (G ) + |λ| · |u|C(G )

(12.18)

N/2p

(A − λ)(ϕ0 (x, η)u) Lp (G )   N/2p N/p |(A − λ)u|C(G ) + η −1 |u|C 1 (G ) + η −2 |u|C(G ) ≤ C |λ| η   ≤ C |λ|N/2p η N/p |(A − λ)u|C(D) + η −1 |u|C 1 (D) + η −2 |u|C(D) ≤ C |λ|

for 0 < η < η0 . We remark (see Figure 12.6 below) that the closure D = D ∪ ∂D can be covered by a finite number of sets of the forms B (x0 , η/2) ∩ D

x0 ∈ ∂D,

and B (x0 , η/2)

x0 ∈ D.

Hence, by taking the supremum of inequality (12.18) over x ∈ D we find that 1/2

|λ|

|u|C 1 (D) + |λ| · |u|C(D)

(12.19)

12.2 Proof of Part (i) of Theorem 1.5

∂D

B(x0 , η/2)

D

355

B(x0 , η/2)

x0 x0

Fig. 12.6. The open ball B (x0 , η/2) in D and the open ball B (x0 , η/2) on ∂D

≤ C |λ|

N/2p

  η N/p |(A − λ)u|C(D) + η −1 |u|C 1 (D) + η −2 |u|C(D)

for all u ∈ D (A). Here we remark that 0 < η < η0 .

(12.20)

(4) We now choose the localization parameter η. To do so, we take a constant K so that (12.21) 0 < K < rp (ε). For a complex number λ = r2 ei θ with r ≥ rp (ε), we let η :=

η0 K, |λ|1/2

(12.22)

where K is chosen later on. Then the parameter η satisfies condition (12.20), since we have, by condition (12.21), η0 η0 η0 K < η0 . K= K≤  η= 1/2 r rp (ε) |λ| Hence it follows from inequality (12.19) that 1/2

|λ|

|u|C 1 (D) + |λ| · |u|C(D)





K −1+N/p |λ| ≤ C η0 K N/p |(A − λ)u|C(D) + C η0   N/p−2 −2+N/p + C η0 K |λ| · |u|C(D) for all u ∈ D (A). N/p

N/p−1

(12.23) 1/2

· |u|C 1 (D)

However, since the exponents −1 + N/p and −2 + N/p are both negative for N < p < ∞, we can choose the constant K so large that N/p−1

C η0 and

K −1+N/p < 1,

356

12 Proof of Theorem 1.5 - Part (i) N/p−2

C η0

K −2+N/p < 1.

For example, we may take   0 := 1 max C 1/σ , C 1/(σ+1) , K>C η0

σ =1−

N > 0. p

(12.24)

Then the desired inequality (12.8) follows from inequality (12.23). Indeed, if we let   0+1 , r(ε) := max rp (ε), C and choose the constant K such that 0 < K < r(ε), C then, for all complex numbers λ = r2 ei θ with r ≥ r(ε) we have, by conditions (12.22) and (12.24), • 0 < η < η0 , • 0 < K < |λ|1/2 , N/p−1

K −1+N/p < 1,

N/p−1

K −1+N/p < 1.

• C η0 • C η0

Now the proof of Lemma 12.2 is complete.   Step (II): The next lemma, together with Lemma 12.2, proves that the resolvent set of A contains the set   Σ(ε) = λ = r2 ei θ : r ≥ r(ε), −π + ε ≤ θ ≤ π − ε , that is, the resolvent (A − λI)−1 exists for all λ ∈ Σ(ε).   Lemma 12.5. If λ ∈ Σ(ε), then, for any f ∈ C0 D \ M , there exists a unique function u ∈ D (A) such that (A − λI)u = f . Proof. Since we have the assertion   f ∈ C0 D \ M ⊂ Lp (D)

for all 1 < p < ∞,

it follows from an application of Theorem 1.4 that if λ ∈ Σ(ε) there exists a unique function u(x) ∈ H 2,p (D) such that (A − λ) u = f and

in D,

  ∂u   Lu = μ(x ) + γ(x )u = 0 on ∂D. ∂n ∂D 

However, part (ii) of Theorem 4.8 asserts that

(12.25)

(12.26)

12.3 Notes and Comments

u ∈ H 2,p (D) ⊂ C 2−N/p (D) ⊂ C 1 (D) if

357

N < p < ∞.

Hence we have, by formula (12.26) and condition (B), u = 0 on M = {x ∈ ∂D : μ(x ) = 0}, so that

  u ∈ C0 D \ M .

Furthermore, in view of formula (12.25) it follows that   Au = f + λu ∈ C0 D \ M . Summing up, we have proved that  u ∈ D (A) , (A − λI)u = f. The proof of Lemma 12.5 is complete.   Now the proof of part (i) of Theorem 1.5 is complete.  

12.3 Notes and Comments This chapter is adapted from Taira [122, Chapter 11], which is an expanded and revised version of Chapter 8 of the second edition [121].

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

In this chapter we prove Theorem 1.6 in Section 13.3 and part (ii) of Theorem 1.5 in Section 13.4. This chapter is the heart of the subject. In Section 13.1 general existence theorems for Feller semigroups are formulated in terms of elliptic boundary value problems with spectral parameter (Theorem 13.14). In Section 13.2 we study Feller semigroups with reflecting barrier (Theorem 13.17) and then, by using these Feller semigroups we construct Feller semigroups corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem 13.22). Our proof is based on the generation theorems of Feller semigroups discussed in Chapter 3. We prove Theorem 1.6 in Section 13.3. To do so, we apply part (ii) of Theorem 3.34 (the Hille–Yosida theorem) to the operator A defined by formula (1.11). The proof of Theorem 1.6 can be flowcharted as in Table 13.1 below. Theorem 3.36 (Hill–Yosida–Ray) Theorem 13.14 (Feller semigroups on ∂D) Lemma 13.11 (operators LHα ) Theorem 1.6 Theorem 13.22 Theorem 3.34 (Hille–Yosida) Theorem 13.17 (Feller semigroups with reflecting barrier) Assertion (13.23) (operators LN Hα )

Table 13.1. A flowchart for the proof of Theorem 1.6

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 13

360

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

The proof of part (ii) of Theorem 1.5 in Section 13.4 can be flowcharted as in Table 13.2 below. density of D( ) in C0 (D \ M ) Theorem 2.2 (analytic semigroups) Part (i) of Theorem 1.5 Part (ii) of Theorem 1.5 Theorem 13.14 (Feller semigroups on ∂D) Theorem 13.22 Theorem 1.6 Theorem 13.17 (Feller semigroups with reflecting barrier)

Table 13.2. A flowchart for the proof of part (ii) of Theorem 1.5

13.1 General Existence Theorem for Feller Semigroups The purpose of this section is to give a general existence theorem for Feller semigroups in terms of boundary value problems (Theorem 13.14), following Taira [122, Chapter 10]. Let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary. We let A=

N  i,j=1

 ∂2 ∂ + bi (x) + c(x) ∂xi ∂xj ∂x i i=1 N

aij (x)

be a second-order, uniformly elliptic differential operator with real coefficients such that: (1) aij ∈ C ∞ (D) and aij (x) = aji (x) for all x ∈ D and 1 ≤ i, j ≤ N , and there exists a positive constant a0 such that N 

aij (x)ξi ξj ≥ a0 |ξ|2

i,j=1

(2) bi ∈ C ∞ (D). (3) c ∈ C ∞ (D) and c(x) ≤ 0 on D.

for all (x, ξ) ∈ D × RN .

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361

The diffusion operator A describes analytically a strong Markov process with continuous paths in the interior D such as Brownian motion (see Figure 1.4). The functions aij (x), bi (x) and c(x) are called the diffusion coefficients, the drift coefficients and the termination coefficient, respectively. Let L be a first-order boundary condition such that Lu = μ(x )

∂u + γ(x )u, ∂n

where: (4) μ ∈ C ∞ (∂D) and μ(x ) ≥ 0 on ∂D. (5) L1 = γ ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit inward normal to the boundary ∂D (see Figure 1.1). The boundary condition L is called a first-order Ventcel’ boundary condition (cf. [143]). Its terms μ(x )(∂u)/(∂n) and γ(x )u are supposed to correspond to the reflection and absorption phenomena, respectively (see Figure 1.3). We are interested in the following problem: Problem. Given analytic data (A, L), can we construct a Feller semigroup {Tt }t≥0 on the state space D whose infinitesimal generator A is characterized by (A, L) ? First, we consider the following Dirichlet problem: Given functions f (x) and ϕ(x ) defined in D and on ∂D, respectively, find a function u(x) in D such that  Au = f in D, (13.1) γ0 u = u|∂D = ϕ on ∂D. The next theorem summarizes the basic facts about the Dirichlet problem in the framework of H¨ older spaces (cf. [50]): Theorem 13.1. (i) (Existence and Uniqueness) If f ∈ C θ (D) with 0 < θ < 1 and if ϕ ∈ C(∂D), then the Dirichlet problem (13.1) has a unique solution u(x) in C(D) ∩ C 2+θ (D). (ii) (Interior Regularity) If u ∈ C 2 (D) and Au = f ∈ C k+θ (D) for some non-negative integer k, then it follows that u ∈ C k+2+θ (D). (iii) (Global Regularity) If f ∈ C k+θ (D) and ϕ ∈ C k+2+θ (∂D) for some non-negative integer k, then a solution u ∈ C(D) ∩ C 2 (D) of the Dirichlet problem (13.1) belongs to the space C k+2+θ (D). In the following we shall use the notation:

f ∞ = max |f (x)| x∈D

|ϕ|∞ = max |ϕ(x )|  x ∈∂D

for f ∈ C(D), for ϕ ∈ C(∂D).

362

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Secondly, we consider the following Dirichlet problem with spectral parameter: For given functions f (x) and ϕ(x ) defined in D and on ∂D, respectively, find a function u(x) in D such that  (α − A) u = f in D, (13.2) γ0 u = u|∂D = ϕ on ∂D, where α is a positive parameter. By applying Theorem 13.1 with A := A − α, we obtain that problem (13.2) has a unique solution u(x) in C 2+θ (D) for any f ∈ C θ (D) and any ϕ ∈ C 2+θ (∂D) with 0 < θ < 1. Therefore, we can introduce linear operators G0α : C θ (D) −→ C 2+θ (D) and Hα : C 2+θ (∂D) −→ C 2+θ (D) as follows. (a) For any f ∈ C θ (D), the function G0α f ∈ C 2+θ (D) is the unique solution of the problem  (α − A) G0α f = f in D,  (13.3) G0α f ∂D = 0 on ∂D. (b) For any ϕ ∈ C 2+θ (∂D), the function Hα ϕ ∈ C 2+θ (D) is the unique solution of the problem  (α − A) Hα ϕ = 0 in D, (13.4) on ∂D. Hα ϕ|∂D = ϕ The operator G0α is called the Green operator and the operator Hα is called the harmonic operator, respectively. Then we have the following lemma: Lemma 13.2. The operator G0α for α > 0, considered from C(D) into itself, is non-negative and continuous (bounded) with norm  0  0  G  = G 1 = max(G0 1)(x). α α α ∞ x∈D

Proof. Let f (x) be an arbitrary function in C θ (D) such that f (x) ≥ 0 on D. Then, by applying the weak maximum principle (see Theorem 10.1) with A := A − α to the function −G0α f we obtain from formula (13.3) that G0α f ≥ 0 on D. This proves the non-negativity of G0α .

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363

Since G0α is non-negative, we have, for all f ∈ C θ (D), −G0α f ∞ ≤ G0α f ≤ G0α f ∞

on D.

This implies the continuity of G0α with norm  0  0  Gα  = Gα 1 . ∞ The proof of Lemma 13.2 is complete.   Similarly, we have the following lemma: Lemma 13.3. The operator Hα for α > 0, considered from C(∂D) into C(D), is non-negative and continuous (bounded) with norm

Hα = Hα 1 ∞ = max(Hα 1)(x). x∈D

More precisely, we have the following fundamental theorem: Theorem 13.4. (i) (a) The operator G0α for α > 0 can be uniquely extended to a non-negative, bounded linear operator on C(D) into itself, denoted again by G0α , with norm  0  0  G  = G 1 ≤ 1 . (13.5) α α ∞ α (b) For any f ∈ C(D), we have the assertion G0α f (x ) = 0

on ∂D.

(c) For all α, β > 0, the resolvent equation holds true: G0α f − G0β f + (α − β)G0α G0β f = 0

for every f ∈ C(D).

(13.6)

(d) For any f ∈ C(D), we have the assertion lim αG0α f (x) = f (x)

α→+∞

for all x ∈ D.

(13.7)

Furthermore, if f (x ) = 0 on ∂D, then this convergence is uniform in x ∈ D, that is, we have the assertion lim αG0α f = f

α→+∞

in C(D).

(13.8)

(e) The operator G0α maps C k+θ (D) into C k+2+θ (D) for any non-negative integer k. (ii) (a ) The operator Hα for α > 0 can be uniquely extended to a nonnegative, bounded linear operator on C(∂D) into C(D), denoted again by Hα , with norm Hα = 1. (b ) For any ϕ ∈ C(∂D), we have the assertion

364

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Hα ϕ|∂D = ϕ

on ∂D.

(c ) For all α, β > 0, we have the equation Hα ϕ − Hβ ϕ + (α − β)G0α Hβ ϕ = 0

for every ϕ ∈ C(∂D).

(13.9)

(d ) The operator Hα maps C k+2+θ (∂D) into C k+2+θ (D) for any non-negative integer k. Proof. (i) (a) By making use of Friedrichs’ mollifiers ([122, Subsection 5.2.6], [136, Subsection 1.3.2]), we find that the H¨ older space C θ (D) is dense in C(D) and further that non-negative functions can be approximated by nonnegative smooth functions. Hence, by Lemma 13.2 it follows that the operator G0α : C θ (D) → C 2+θ (D) can be uniquely extended to a non-negative, bounded linear operator G0α : C(D) −→ C(D)  0   0 with norm Gα  = Gα 1∞ . Furthermore, since the function G0α 1 satisfies the conditions  (A − α) G0α 1 = −1 in D,  G0α 1∂D = 0 on ∂D, by applying Theorem 10.2 with A := A − α we obtain that  0  0  Gα  = Gα 1 ≤ 1 . ∞ α (b) This assertion follows from formula (13.3), since the space C θ (D) is dense in C(D) and since the operator G0α : C(D) → C(D) is bounded. (c) We find from the uniqueness theorem for problem (13.2) (Theorem 13.1) that equation (13.6) holds true for all f ∈ C θ (D). Indeed, it suffices to note that the function v := G0α f − G0β f + (α − β) G0α G0β f ∈ C 2+θ (D) satisfies the conditions



(α − A) v = 0 v|∂D = 0

in D, on ∂D,

so that v = 0 in D. Therefore, we obtain that the resolvent equation (13.6) holds true for all f ∈ C(D), since the space C θ (D) is dense in C(D) and since the operators G0α and G0β are bounded. (d) First, let f (x) be an arbitrary function in C θ (D) satisfying the boundary condition f |∂D = 0. Then it follows from an application of the uniqueness theorem for problem (13.2) (Theorem 13.1) that we have, for all α, β,

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365

f − αG0α f = G0α ((β − A)f ) − βG0α f. Indeed, the both sides satisfy the same equation (α − A)u = −Af in D and have the same boundary value 0 on ∂D. Thus we have, by estimate (13.5),   f − αG0α f  ≤ 1 (β − A)f + β f , ∞ ∞ ∞ α α so that

  lim f − αG0α f ∞ = 0.

(13.10)

α→+∞

Now let f (x) be an arbitrary function in C(D) satisfying the boundary condition f |∂D = 0. By means of mollifiers ([122, Subsection 5.2.6], [136, ∞ Subsection 1.3.2]), we can find a sequence {fj }j=1 in C θ (D) such that 

fj −→ f fj |∂D = 0

in C(D) as j → ∞, on ∂D.

Then we have, by estimate (13.5) and assertion (13.10) with f := fj ,       f − αG0α f  ≤ f − fj + fj − αG0α fj  + αG0α (fj − f ) ∞ ∞ ∞ ∞   ≤ 2 f − fj + fj − αG0 fj  , ∞

and hence

α

∞

  lim sup f − αG0α f ∞ ≤ 2 f − fj ∞ . α→+∞

This proves the desired assertion (13.8), since f − fj ∞ → 0 as j → ∞. To prove assertion (13.7), let f (x) be an arbitrary function in C(D) and let x be an arbitrary point of D. If we take a function ψ(y) in C(D) such that ⎧ ⎪ ⎨0 ≤ ψ(y) ≤ 1 on D, ψ(y) = 0 in a neighborhood of x, ⎪ ⎩ ψ(y) = 1 near the set ∂D, then it follows from the non-negativity of G0α and estimate (13.5) that 0 ≤ αG0α ψ(x) + αG0α (1 − ψ)(x) = αG0α 1(x) ≤ 1.

(13.11)

However, by applying assertion (13.8) to the function 1 − ψ(y) we have the assertion lim αG0α (1 − ψ)(x) = (1 − ψ)(x) = 1

α→+∞

for all x ∈ D.

In view of inequalities (13.11), this implies that lim αG0α ψ(x) = 0 for all x ∈ D.

α→+∞

366

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Thus, since we have the inequalities − f ∞ ψ(x) ≤ f (x)ψ(x) ≤ f ∞ ψ(x) it follows that, for x ∈ D,  0  αGα (f ψ)(x) ≤ f · αG0α ψ(x) −→ 0 ∞

on D,

as α → +∞.

Therefore, by applying assertion (13.8) to the function (1 − ψ(y))f (y) we obtain that f (x) = ((1 − ψ)f ) (x) = lim αG0α ((1 − ψ)f ) (x) α→+∞

= lim αG0α f (x) for all x ∈ D. α→+∞

This proves the desired assertion (13.7). (ii) (a ) Since the space C 2+θ (∂D) is dense in C(∂D), by Lemma 13.3 it follows that the operator Hα : C 2+θ (∂D) → C 2+θ (D) can be uniquely extended to a non-negative, bounded linear operator Hα : C(∂D) −→ C(D). Furthermore, by applying Theorem 10.2 with A := A−α we have the assertion

Hα = Hα 1 ∞ = 1. (b ) This assertion follows from formula (13.4), since C 2+θ (∂D) is dense in C(∂D) and since the operator Hα : C(∂D) → C(D) is bounded. (c ) We find from the uniqueness theorem for problem (13.2) that equation (13.9) holds true for all ϕ ∈ C 2+θ (∂D). Indeed, it suffices to note that the function w := Hα ϕ − Hβ ϕ + (α − β)G0α Hβ ϕ ∈ C 2+θ (D) satisfies the conditions



(α − A) w = 0 w|∂D = 0

in D, on ∂D,

so that w = 0 in D. Therefore, we obtain that the desired equation (13.9) holds true for all ϕ ∈ C(∂D), since the space C 2+θ (∂D) is dense in C(∂D) and since the operators G0α and Hα are bounded. The proof of Theorem 13.4 is now complete.   Summing up, we have the mapping properties of the operators G0α and Hα as in Figures 13.1 and 13.2 below.

13.1 General Existence Theorem for Feller Semigroups C(D)

G0

α −−−− − →

367

C(D)

→ C 2+θ (D) D(G0α ) = C θ (D) −−−−− G0 α

Fig. 13.1. The mapping properties of the operator G0α C(∂D)

H

α − −−− − →

C(D)

−−−− → C 2+θ (D) D(Hα ) = C 2+θ (∂D) − Hα

Fig. 13.2. The mapping properties of the operator Hα

Now we consider the following Ventcel’ boundary value problem in the framework of the spaces of continuous functions.  (α − A) u = f in D, (13.12) Lu = 0 on ∂D. To do this, we introduce three operators associated with the Ventcel’ boundary value problem (13.12). Step (I): First, we introduce a linear operator A : C(D) −→ C(D) as follows. (a) The domain D(A) of A is the space C 2 (D). &N &N ∂2u ∂u + i=1 bi (x) + c(x)u, u ∈ D(A). (b) Au = i,j=1 aij (x) ∂xi ∂xj ∂xi Then we have the following lemma: Lemma 13.5. The operator A has its minimal closed extension A in the space C(D). Proof. We apply part (i) of Theorem 3.36 (the Hille–Yosida–Ray theorem) to the operator A. Assume that a function u ∈ C 2 (D) takes a positive maximum at an interior point x0 of D:

368

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

u(x0 ) = max u(x) > 0. x∈D

Then it follows that ∂u (x0 ) = 0 for 1 ≤ i ≤ N , ∂xi N  ∂2u • aij (x0 ) (x0 ) ≤ 0, ∂xi ∂xj i,j=1 •

since the matrix (aij (x)) is positive definite. Hence we have the assertion Au(x0 ) =

N 

aij (x0 )

i,j=1

∂2u (x0 ) + c(x0 )u(x0 ) ≤ 0, ∂xi ∂xj

This implies that the operator A satisfies condition (β) of Theorem 3.36 with K0 := D and K := D. Therefore, Lemma 13.5 follows from an application of the same theorem. The proof of Lemma 13.5 is complete.   Remark 13.6. Since the injection: C(D) → D (D) is continuous, we have the formula Au =

N  i,j=1

 ∂2u ∂u + bi (x) + c(x)u ∂xi ∂xj ∂x i i=1 N

aij (x)

for every u ∈ C(D),

where the right-hand side is taken in the sense of distributions. The operators A and A can be visualized as in Figure 13.3 below. C(D)

D(A)

A

−−−−− → C(D)

→ C(D) D(A) = C 2 (D) −−−−− A

Fig. 13.3. The operators A and A

The extended operators G0α : C(D) −→ C(D), Hα : C(∂D) −→ C(D) still satisfy formulas (13.3) and (13.4) respectively in the following sense:

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369

Lemma 13.7. Let α > 0. Then we have the following two assertions (i) and (ii): (i) For any f ∈ C(D), we have the formulas    G0α f ∈ D A ,   αI − A G0α f = f in D. (ii) For any ϕ ∈ C(∂D), we have the formulas    Hα ϕ ∈ D A ,   αI − A Hα ϕ = 0 in D.   Here D A is the domain of the closed extension A.

Proof. (i) By making use of Friedrichs’ mollifiers ([122, Subsection 5.2.6], [136, θ Subsection 1.3.2]), we can choose a sequence {fj }∞ j=1 in C (D) such that fj → f in C(D) as j → ∞. Then it follows from the boundedness of G0α that G0α fj −→ G0α f

in C(D),

and further that (α − A) G0α fj = fj −→ f Hence we have the assertions    G0α f ∈ D A ,   αI − A G0α f = f

in C(D).

in D.

since the operator A : C(D) → C(D) is closed. (ii) Similarly, part (ii) is proved, since the space C 2+θ (∂D) is dense in C(∂D) and since the operator Hα : C(∂D) → C(D) is bounded. The proof of Lemma 13.7 is complete.   Then we have the following corollary:   Corollary 13.8. Every function u in D A can be written in the following form:   (13.13) u = G0α (αI − A)u + Hα (u|∂D ) for all α > 0. Proof. We let

  w := u − G0α (αI − A)u − Hα (u|∂D ).

  Then it follows from Lemma 13.7 that the function w is in D A and satisfies the conditions

370

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

  αI − A w = 0 in D, w|∂D = 0 on ∂D. However, in light of Remark 13.6, by applying Lemma 7.1 (cf. [68, Theorem 3.1], [123, Theorem 7.1]) and Theorem 6.22 with A := A − α to the Dirichlet case (μ(x ) ≡ 0 and γ(x ) ≡ −1 on ∂D) we obtain that w ∈ C ∞ (D). Therefore, it follows from an application of part (i) of Theorem 13.1 with A := A − α that w = 0. This proves the desired formula (13.13). The proof of Corollary 13.8 is complete.   Therefore, we can express the relationships between the operators A, G0α and Hα in matrix form:      I 0 αI − A  0 Gα Hα = . 0 I γ0 Step (II): Secondly, we introduce a linear operator LG0α : C(D) −→ C(∂D) as follows.

 0 0 θ (a) The domain D LG older  space C (D) with 0 < θ < 1.  α of LGα is the  H¨ 0 0 0 (b) LGα f = L Gα f for every f ∈ D LGα . Then we have the following lemma: Lemma 13.9. The operator LG0α for α > 0 can be uniquely extended to a non-negative, bounded linear operator LG0α : C(D) → C(∂D).   Proof. Let f (x) be an arbitrary function in D LG0α = C θ (D) such that f (x) ≥ 0 on D. Then we have the assertions ⎧ 0 2+θ ⎪ (D), ⎨Gα f ∈ C G0α f ≥ 0 on D, ⎪ ⎩ 0  on ∂D, Gα f ∂D = 0 and hence LG0α f

  ∂ 0  0 = μ(x ) (Gα f ) + γ(x )(Gα f ) ∂n ∂D ∂  0 (G f ) ≥ 0 on ∂D. = μ(x ) ∂n α 

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371

This proves that the operator LG0α is non-negative.   By the non-negativity of LG0α , we have, for all f ∈ D LG0α , −LG0α f ∞ ≤ LG0α f ≤ LG0α f ∞

on ∂D.

This implies the boundedness of LG0α with norm  0  0  LGα  = LGα 1 = max (LG0α 1)(x ). ∞  x ∈∂D

Recall that the space C θ (D) is dense in C(D) and that non-negative functions can be approximated by non-negative smooth functions. Hence we find that the operator LG0α can be uniquely extended to a non-negative, bounded linear operator LG0α : C(D) → C(∂D). The proof of Lemma 13.9 is complete.   The operators LG0α and LG0α can be visualized in Figure 13.4 below. LG0

D(LG0α ) = C(D) − −−−α− →

C(∂D)

−−−− → C 1+θ (∂D) D(LG0α ) = C θ (D) − LG0 α

Fig. 13.4. The operators LG0α and LG0α

The next lemma states a fundamental relationship between the operators LG0α and LG0β for α, β > 0: Lemma 13.10. For any f ∈ C(D), we have the formula LG0α f − LG0β f + (α − β) LG0α G0β f = 0

for all α, β > 0.

(13.14)

Proof. Choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞, just as in Lemma 13.7. Then, by using the resolvent equation (13.6) with f := fj we have the formula LG0α fj − LG0β fj + (α − β)LG0α G0β fj = 0. Hence, the desired formula (13.14) follows by letting j → ∞, since the operators LG0α , LG0β and G0β are all bounded. The proof of Lemma 13.10 is complete.   Step (III): Finally, we introduce a linear operator LHα : C(∂D) −→ C(∂D) as follows.

372

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

(a) The domain D (LHα ) of LHα is the space C 2+θ (∂D). (b) LHα ψ = L (Hα ψ) for every ψ ∈ D (LHα ). Then we have the following lemma: Lemma 13.11. The operator LHα for α > 0 has its minimal closed extension LHα in the space C(∂D). Proof. We apply part (i) of Theorem 3.36 (the Hille–Yosida–Ray theorem) to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (β  ) with K := ∂D (or condition (β) with K := K0 = ∂D) of the same theorem. Assume that a function ψ in D (LHα ) = C 2+θ (∂D) takes its positive maximum at some point x of ∂D. Since the function Hα ψ is in C 2+θ (D) and satisfies  (A − α) Hα ψ = 0 in D, Hα ψ|∂D = ψ on ∂D, by applying the weak maximum principle (Theorem 10.1) with A := A − α to the function Hα ψ we find that the function Hα ψ takes its positive maximum at the boundary point x ∈ ∂D. Thus we can apply the Hopf boundary point lemma (Lemma 10.11) with A := A − α to obtain that ∂ (Hα ψ)(x ) < 0. ∂n Hence we have the inequality LHα ψ(x ) =

N −1 

αij (x )

i,j=1

∂2ψ ∂ (x ) + μ(x ) (Hα ψ)(x ) + γ(x )ψ(x ) ∂xi ∂xj ∂n

≤ 0. This verifies condition (β  ) of Theorem 3.36. Therefore, Lemma 13.11 follows from an application of the same theorem. The proof of Lemma 13.11 is complete.   The operators LHα and LHα can be visualized in Figure 13.5 below. Remark 13.12. The operator LHα enjoys the following property:   If a function ψ ∈ D LHα takes its positive maximum

(13.15)



at some point x of ∂D, then we have the inequality LHα ψ(x ) ≤ 0. The next lemma states a fundamental relationship between the operators LHα and LHβ for α, β > 0:

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373

C(∂D)

D(LHα )

LH

− −−−α −→ C(∂D)

−−−−→ C θ (∂D) D(LHα ) = C 2+θ (∂D) − LHα

Fig. 13.5. The operators LHα and LHα

  Lemma 13.13. The domain D LHα of LHα does not depend on α > 0; so we denote by D the common domain. Then we have the formula LHα ψ − LHβ ψ + (α − β)LG0α Hβ ψ = 0

(13.16)

for all α, β > 0 and ψ ∈ D.   Proof. Let ψ(x ) be an arbitrary function in D LHβ , and choose a sequence {ψj } in D (LHβ ) = C 2+θ (∂D) such that, as j → ∞,  ψj −→ ψ in C(∂D), LHβ ψj −→ LHβ ψ in C(∂D). Then it follows from the boundedness of Hβ and LG0α that LG0α (Hβ ψj ) = LG0α (Hβ ψj ) −→ LG0α (Hβ ψ)

in C(∂D).

Therefore, by using formula (13.9) with ϕ := ψj we obtain that, as j → ∞, LHα ψj = LHβ ψj − (α − β)LG0α (Hβ ψj ) −→ LHβ ψ − (α − β)LG0α (Hβ ψ) in C(∂D). This implies that 

  ψ ∈ D LHα , LHα ψ = LHβ ψ − (α − β)LG0α (Hβ ψ),

since the operator LHα : C(∂D) → C(∂D) is closed. Conversely, by interchanging α and β we have the assertion     D LHα ⊂ D LHβ , and so

    D LHα = D LHβ .

The proof of Lemma 13.13 is complete.  

374

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Now we can prove a general existence theorem for Feller semigroups on the state space ∂D in terms of boundary value problem (13.12). The next theorem asserts that the operator LHα is the infinitesimal generator of some Feller semigroup on the state space ∂D if and only if problem (13.12) is solvable for sufficiently many functions ϕ in the Banach space C(∂D): Theorem 13.14. (i) If the operator LHα erator of a Feller semigroup on the state constant λ, the boundary value problem  (α − A) u = 0 (λ − L) u = ϕ

for α > 0 is the infinitesimal genspace ∂D, then, for each positive

in D, on ∂D

(13.17)

has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D). (ii) Conversely, if, for some non-negative constant λ, problem (13.17) has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D), then the operator LHα is the infinitesimal generator of some Feller semigroup on the state space ∂D. Proof. (i) If the operator LHα generates a Feller semigroup on the state space ∂D, by applying part (i) of Theorem 3.36 (the Hille–Yosida–Ray theorem) with K := ∂D to the operator LHα we obtain that   R λI − LHα = C(∂D) for each λ > 0. This implies that the range R (λI − LHα ) is a dense subset of C(∂D) for each λ > 0. However, if ϕ ∈ C(∂D) is in the range R (λI − LHα ) and if ϕ = (λI − LHα ) ψ with ψ ∈ C 2+θ (∂D), then the function u = Hα ψ ∈ C 2+θ (D) is a solution of problem (13.17). This proves part (i) of Theorem 13.14. (ii) We apply part (ii) of Theorem 3.36 with K := ∂D to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (γ) of the same theorem, since it satisfies condition (β  ), as is shown in the proof of Lemma 13.11. By the uniqueness theorem for problem (13.2), it follows that any function u ∈ C 2+θ (D) which satisfies the homogeneous equation (α − A) u = 0

in D

can be written in the form: u = Hα (u|∂D ) ,

u|∂D ∈ C 2+θ (∂D) = D (LHα ) .

Thus we find that if there exists a solution u ∈ C 2+θ (D) of problem (13.17) for a function ϕ ∈ C(∂D), then we have the assertion (λI − LHα ) (u|∂D ) = ϕ, and so

13.1 General Existence Theorem for Feller Semigroups

375

ϕ ∈ R (λI − LHα ) . Hence, if, for some non-negative constant λ, problem (13.12) has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D), then the range R (λI − LHα ) is dense in C(∂D). This verifies condition (γ) (with α0 := λ) of Theorem 3.36. Therefore, part (ii) of Theorem 13.14 follows from an application of the same theorem. Now the proof of Theorem 13.14 is complete.   Remark 13.15. Intuitively, Theorem 13.14 asserts that we can “piece together” a Markov process (Feller semigroup) on the boundary ∂D with A-diffusion in the interior D to construct a Markov process (Feller semigroup) on the closure D = D ∪ ∂D. The situation may be represented schematically as in Figure 13.6 below.

∂D

D

Fig. 13.6. A Markov process on ∂D pieced together with an A-diffusion in D

We conclude this section by giving   a precise meaning to the boundary conditions Lu for functions u in D A . We let     D(L) = u ∈ D A : u|∂D ∈ D , where D is the common domain of the operators LHα for all α > 0. We remark that the space D(L) contains C 2+θ (D), since C 2+θ (∂D) = D (LHα ) ⊂ D. Corollary 13.8 asserts that every function u in D(L) ⊂ D A can be written in the form   (13.13) u = G0α (αI − A)u + Hα (u|∂D ) for all α > 0. Then we define the boundary condition Lu by the formula   Lu = LG0α (αI − A)u + LHα (u|∂D ) for every u ∈ D(L).

(13.18)

The next lemma justifies definition (13.18) of Lu for each u ∈ D(L):

376

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Lemma 13.16. The right-hand side of formula (13.18) depends only on u, not on the choice of expression (13.13). Proof. Assume that    αI − A u + Hα (u|∂D )    = G0β βI − A u + Hβ (u|∂D ) ,

u = G0α

where α, β > 0. Then it follows from formula (13.14) with   f := αI − A u and formula (13.18) with ψ := u|∂D that    αI − A u + LHα (u|∂D )       = LG0β αI − A u − (α − β)LG0α G0β αI − A u LG0α

(13.19)

+ LHβ (u|∂D ) − (α − β)LG0α Hβ (u|∂D ) = LG0β ((βI − A)u) + LHβ (u|∂D )     + (α − β) LG0β u − LG0α G0β αI − A u − LG0α Hβ (u|∂D ) . However, the last term of formula (13.19) vanishes. Indeed, it follows from formula (13.13) with α := β and formula (13.14) with f := u that     LG0β u − LG0α G0β αI − A u − LG0α Hβ (u|∂D )     = LG0β u − LG0α G0β βI − A u + Hβ (u|∂D ) + (α − β)G0β u = LG0β u − LG0α u − (α − β)LG0α G0β u = 0. Therefore, we obtain from formula (13.19) that       LG0α αI − A u + LHα (u|∂D ) = LG0β βI − A u + LHβ (u|∂D ) . The proof of Lemma 13.16 is complete.  

13.2 Feller Semigroups with Reflecting Barrier Now we consider the Neumann boundary condition  ∂u  . LN u ≡ ∂n ∂D

13.2 Feller Semigroups with Reflecting Barrier

377

We recall that the boundary condition LN is supposed to correspond to the reflection phenomenon. The next theorem (formula (13.21)) asserts that we can “piece together” a Markov process on ∂D with A-diffusion in D to construct a Markov process on D = D ∪ ∂D with reflecting barrier (cf. [15, Th´eor`eme XIX]): Theorem 13.17 (the reflecting diffusion). We define a linear operator AN : C(D) −→ C(D) as follows. (a) The domain D (AN ) of AN is the space     D (AN ) = u ∈ D A : u|∂D ∈ DN , LN u = 0 ,

(13.20)

where DN is the common domain of the operators LN Hα for all α > 0. (b) AN u = Au for every u ∈ D (AN ). Then the operator AN is the infinitesimal generator of some Feller semi−1 for group on the state space D, and the Green operator GN α = (αI − AN ) α > 0 is given by the following formula:    −1 0 0f f = G f − H L H L G for every f ∈ C(D). (13.21) GN α N α N α α α Remark 13.18. (1) We can express the relationships between the operators A, GN α , Hα and LN Hα in matrix form:     I 0 −1 αI − A GN = . H L H α N α α 0 I LN (2) In terms of the Boutet de Monvel calculus, we can express the Green operator GN α in the matrix form via the Boutet de Monvel calculus (see Figure 13.7 below) ⎞ ⎛ 0 −Hα Gα C(D) C(D) ⎠: ⎝ −→ −1 C(∂D) C(∂D). LN G0α LN Hα Proof. In order to prove Theorem 13.17, we apply part (ii) of Theorem 3.34 (the Hille–Yosida theorem) to the operator AN defined by formula (14.20). The proof is divided into eight steps. Step 1: First, we prove that: The operator LN Hα is the generator of some Feller semigroup, on the state space ∂D for any sufficiently large α > 0.

378

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6 LN G0

α C(D) − −−−− →

G0 α

C(∂D) LN Hα

−1

C(D) ←−−−− − DN = D(LN Hα ) −Hα

Fig. 13.7. The mapping properties of the terms of the Green operator GN α defined by formula (13.21)

We introduce a linear operator TN (α) = LN Hα : B 1−1/p,p (∂D) −→ B 1−1/p,p (∂D) for N < p < ∞ as follows. (a) The domain D (TN (α)) of TN (α) is the space   D (TN (α)) = ϕ ∈ B 1−1/p,p (∂D) : LN Hα ϕ ∈ B 1−1/p,p (∂D) . (b) TN (α)ϕ = LN Hα ϕ for every ϕ ∈ D (TN (α)). Here it should be emphasized that the harmonic operator Hα is essentially the same as the Poisson operator P (α) introduced in [121, p. 89, Section 5.2]. Then, by arguing just as in the proof of [121, p. 101, Theorem 7.1] with μ(x ) := 1 and γ(x ) := 0 on ∂D we obtain that The operator TN (α) is bijective for all sufficiently large α > 0. Furthermore, TN (α) maps the space C ∞ (∂D) onto itself.

(13.22)

Since we have the assertion LN Hα = TN (α)

on C ∞ (∂D),

it follows from assertion (13.22) that the operator LN Hα also maps C ∞ (∂D) onto itself, for any sufficiently large α > 0. This implies that the range R(LN Hα ) is a dense subset of C(∂D). Hence, by applying part (ii) of Theorem 13.14 we obtain that the operator LN Hα generates a Feller semigroup on the state space ∂D, for any sufficiently large α > 0. The situation can be visualized in Figure 13.8 below. Step 2: Next we prove that: The operator LN Hβ generates a Feller semigroup, on the state space ∂D for any β > 0.

13.2 Feller Semigroups with Reflecting Barrier DN = D(LN Hα )

D(TN (α))

L

Hα

N −−− −− →

379

C(∂D)

TN (α)

−−−−− → B 1−1/p,p (∂D)

→ D(LN Hα ) = C ∞ (∂D) −−−−− LN Hα

C ∞ (∂D)

Fig. 13.8. The mapping properties of the operators TN (α) and LN Hα for N < p < ∞

We take a positive constant α so large that the operator LN Hα generates a Feller semigroup on the state space ∂D. We apply Corollary 3.37 with K := ∂D to the operator LN Hβ for β > 0. By formula (13.16), it follows that the operator LN Hβ can be written as LN Hβ = LN Hα + Nαβ , where Nαβ = (α − β) LN G0α Hβ is a bounded linear operator on C(∂D) into itself. Furthermore, assertion (13.16) implies that the operator LN Hβ satisfies condition (β  ) of Theorem 3.36 (the Hille–Yosida–Ray theorem). Therefore, it follows from an application of Corollary 3.37 that the operator LN Hβ also generates a Feller semigroup on the state space ∂D. Step 3: Now we prove that: The equation

(13.23)

L N Hα ψ = ϕ

  has a unique solution ψ in D LN Hα for any ϕ ∈ C(∂D); hence the inverse LN Hα

−1

of LN Hα can be defined on the whole space C(∂D).

Furthermore, the operator −LN Hα

−1

is non-negative and bounded

on the space C(∂D). Since the function Hα 1(x) takes its positive maximum 1 only on the boundary ∂D, we can apply the Hopf boundary point lemma (Lemma 10.11) to obtain that ∂ (Hα 1) < 0 on ∂D. (13.24) ∂n Hence the Neumann boundary condition implies that

380

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

LN Hα 1 = LN (Hα 1) = and so

∂ (Hα 1) < 0 on ∂D, ∂n

α = min (−LN Hα 1)(x ) = − max (LN Hα 1)(x ) > 0.   x ∈∂D

x ∈∂D

Furthermore, by using Corollary 3.35 with ⎧ ⎪ ⎨K := ∂D, A := LN Hα , ⎪ ⎩ c := α , we obtain that the operator LN Hα + α I is the infinitesimal generator of some Feller semigroup on the state space ∂D. Therefore, since α > 0, it follows from an application of part (i) of Theorem 3.34 (the Hille–Yosida theorem) with A := LN Hα + α I that the equation   −LN Hα ψ = α I − (LN Hα + α I) ψ = ϕ   has a unique solution ψ ∈ D LN Hα for any ϕ ∈ C(∂D), and further that the  −1 −1 is non-negative and bounded operator −LN Hα = α I − (LN Hα + α I) on the space C(∂D) with norm    −1  1 −1     . ≤ −LN Hα  =  α I − (LN Hα + α I) α Step 4: By assertion (13.23), we can define the right-hand side of formula (13.21) for all α > 0. We prove that: GN α = (αI − AN )

−1

for all α > 0.

(13.25)

In view of Lemmas 13.7 and 13.16 with L := LN , it follows that we have, for all f ∈ C(D), ⎧      −1 N 0 0f ⎪ G f = G f − H L H L G ∈D A , ⎪ α N α N α α α ⎪ ⎨     −1 GN LN G0α f ∈ D LN Hα = DN , α f |∂D = −LN Hα ⎪    ⎪ −1 ⎪ 0 ⎩LN (GN LN G0α f = 0, α f ) = LN Gα f − LN Hα LN Hα and (αI − A)(GN α f ) = f. Hence we have proved that, for all f ∈ C(D),  GN α f ∈ D (AN ) , (αI − AN ) GN α f = f.

13.2 Feller Semigroups with Reflecting Barrier

381

This proves that (αI − AN ) GN α = I

on C(D).

Therefore, in order to prove formula (13.25) it suffices to show the injectivity of the operator αI − AN for α > 0. Assume that: u ∈ D (AN ) and (αI − AN ) u = 0. Then, by Corollary 13.8 it follows that the function u can be written as   u = Hα (u|∂D ) , u|∂D ∈ DN = D LN Hα . Thus we have the assertion LN Hα (u|∂D ) = LN u = 0. In view of assertion (13.23), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0 in D. This proves the injectivity of αI − AN for α > 0. Step 5: The non-negativity of GN α for α > 0 follows immediately from −1 and LN G0α are all formula (13.21), since the operators G0α , Hα , −LN Hα non-negative. Step 6: We prove that the operator GN α is bounded on the space C(D) with norm  N G  ≤ 1 for all α > 0. (13.26) α α To do this, it suffices to show that, for all α > 0, GN α1≤

1 α

on D.

(13.27)

since GN α is non-negative on C(D). First, it follows from the uniqueness property of solutions of problem (13.2) that (13.28) αG0α 1 + Hα 1 = 1 + G0α (c(x)) on D. Indeed, the both sides satisfy the same equation (α − A)u = α in D and have the same boundary value 1 on ∂D. By applying the boundary operator LN to the both hand sides of equality (13.28), we obtain that −LN Hα 1 = −LN (Hα 1) = −LN 1 − LN G0α (c(x)) + αLN G0α 1

382

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

  ∂  0 Gα (c(x))  + αLN G0α 1 =− ∂n ∂D  on ∂D, ≥ αLN G0 1 α

∂D

 since G0α (c(x))∂D = 0 on ∂D and G0α (c(x)) ≤ 0 on D. Hence we have, by the non-negativity of −LN Hα

−1

,

− L N Hα

−1

  1 LN G0α 1 ≤ α

on ∂D.

(13.29)

By using formula (13.21) with f := 1, inequality (13.29) and equality (13.28), we obtain that   −1  0 GN LN G0α 1 α 1 = Gα 1 + Hα −LN Hα ≤ G0α 1 + ≤

1 α

1 1 1 Hα 1 = + G0α (c(x)) α α α

on D,

since the operators Hα and G0α are non-negative and since G0α (c(x)) ≤ 0 on D. Therefore, we have proved the desired assertion (13.27) for all α > 0. Step 7: Finally, we prove that: The domain D (AN ) is dense in the space C(D).

(13.30)

Step 7-1: Before the proof, we need some lemmas on the behavior of G0α , −1 Hα and −LN Hα as α → +∞: Lemma 13.19. For all f ∈ C(D), we have the assertion   lim αG0α f + Hα (f |∂D ) = f in C(D). α→+∞

(13.31)

Proof. Choose a positive constant β and let g := f − Hβ (f |∂D ). Then, by using formula (13.9) with ϕ := f |∂D we obtain that   αG0α g − g = αG0α f + Hα (f |∂D ) − f − βG0α Hβ (f |∂D ). However, we have, by estimate (13.5), lim G0α Hβ (f |∂D ) = 0 in C(D),

α→+∞

and, by assertion (13.8),

(13.32)

13.2 Feller Semigroups with Reflecting Barrier

lim αG0α g = g

α→+∞

383

in C(D),

since g|∂D = 0. Therefore, the desired formula (13.31) follows by letting α → +∞ in formula (13.32). The proof of Lemma 13.19 is complete.   Lemma 13.20. The function   ∂ (Hα 1) ∂n ∂D diverges to −∞ uniformly and monotonically as α → +∞. Proof. First, formula (13.9) with ϕ := 1 gives that Hα 1 = Hβ 1 − (α − β)G0α Hβ 1. Thus, in view of the non-negativity of G0α and Hα it follows that α ≥ β =⇒ Hα 1 ≤ Hβ 1

on D.

Since Hα 1|∂D = Hβ 1|∂D = 1, this implies that the functions   ∂ (Hα 1) ∂n ∂D are monotonically non-increasing in α. Furthermore, by using formula (13.7) with f := Hβ 1 we find that the function   β Hα 1(x) = Hβ 1(x) − 1 − αG0α Hβ 1(x) α converges to zero monotonically as α → +∞, for each interior point x of D. Now, for any given positive constant K we can construct a function u ∈ C 2 (D) such that u|∂D = 1 on ∂D,  ∂u  ≤ −K on ∂D. ∂n 

(13.33a) (13.33b)

∂D

Indeed, it follows from an application of Theorem 13.1 that, for any integer m > 0, the function m u = (Hα0 1) belongs to C 2+θ (D) and satisfies condition (13.33a). Furthermore, we have the assertion

384

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

   ∂u  ∂ (Hα0 1) =m·  ∂n ∂D ∂n ∂D   ∂  (H ≤ m · max 1) (x ) . α0 x ∈∂D ∂n In view of inequality (13.24) with α := α0 , this implies that the function m u = (Hα0 u) satisfies condition (13.33b) for m sufficiently large. We take a function u ∈ C 2 (D) which satisfies conditions (13.33a) and (13.33b), and choose a neighborhood U of ∂D, relative to D, with smooth boundary ∂U such that (see Figure 13.9 above) u≥

1 2

on U .

(13.34)

∂D U U

U ∂D

∂D

Fig. 13.9. The neighborhood U of ∂D

Recall that the function Hα 1 converges to zero in the interior D monotonically as α → +∞. Since u = Hα 1 = 1 on the boundary ∂D, by using Dini’s theorem we can find a positive constant α (depending on u and hence on K) such that Hα 1 ≤ u on ∂U \ ∂D, α > 2 Au ∞ . It follows from inequalities (13.34) and (13.35b) that (A − α) (Hα 1 − u) = αu − Au ≥ > 0 in U .

α − Au ∞ 2

(13.35a) (13.35b)

13.2 Feller Semigroups with Reflecting Barrier

385

Thus, by applying the weak maximum principle (Theorem 10.1) with A := A − α to the function Hα 1 − u we obtain that the function Hα 1 − u may take its positive maximum only on the boundary ∂U . However, conditions (13.33a) and (13.35a) imply that Hα 1 − u ≤ 0 on ∂U = (∂U \ ∂D) ∪ ∂D. Therefore, we have the assertion Hα 1 ≤ u and hence

on U = U ∪ ∂U ,

   ∂ ∂u   (Hα 1) ≤ ≤ −K ∂n ∂n ∂D ∂D

on ∂D,

since u|∂D = Hα 1|∂D = 1. The proof of Lemma 13.20 is complete.   Now we can study the behavior of the operator norm − LN Hα α → +∞: Corollary 13.21. limα→+∞ − LN Hα

−1

−1

as

= 0.

Proof. By Lemma 13.20, it follows that the function LN Hα 1(x ) = LN (Hα 1) (x ) =

∂ (Hα 1) (x ) for x ∈ ∂D, ∂n

diverges to −∞ monotonically as α → +∞. By Dini’s theorem, this convergence is uniform in x ∈ ∂D. Hence the function 1 LN Hα 1(x ) converges to zero uniformly in x ∈ ∂D as α → +∞. This implies that     −1  −1    −LN Hα  = −LN Hα 1 ∞     1  −→ 0 as α → +∞. ≤  LN Hα 1 ∞ Indeed, it suffices to note that    −LN Hα 1(x )  1  (−LN Hα 1(x )) ≤ 1=   |LN Hα 1(x )| L N Hα 1  ∞ The proof of Corollary 13.21 is complete.  

for all x ∈ ∂D.

386

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Step 7-2: Proof of Assertion (13.30) In view of formula (13.25) and inequality (13.26), it suffices to prove that   ∞  (13.36) lim αGN α f − f ∞ = 0 for every f ∈ C (D), α→+∞

since the space C ∞ (D) is dense in C(D). First, we remark that      N  −1   0 0 αGα f − f  =  αG L f − αH L H G f − f   α N α N α α ∞ ∞  0  ≤ αGα f + Hα (f |∂D ) − f ∞     −1    LN G0α f − Hα (f |∂D ) + −αHα LN Hα ∞  0    ≤ αGα f + Hα (f |∂D ) − f ∞    −1    LN G0α f − (f |∂D ) . + −αLN Hα ∞

Thus, in view of assertion (13.31) it suffices to show that C D  −1  lim −αLN Hα LN G0α f − (f |∂D ) = 0 in C(∂D). α→+∞

(13.37)

We take a constant β such that 0 < β < α, and write f = G0β g + Hβ ϕ, where (cf. formula (13.13)):  g = (β − A)f ∈ C θ (D), ϕ = f |∂D ∈ C 2+θ (∂D). Then, by using equations (13.6) (with f := g) and (13.9) we obtain that G0α f = G0α G0β g + G0α Hβ ϕ  1  0 = Gβ g − G0α g + Hβ ϕ − Hα ϕ . α−β Hence we have the assertion    −1    LN G0α f − (f |∂D ) −αLN Hα ∞      α   α −1  0 0  LN Gβ g − LN Gα g + LN Hβ ϕ + −LN Hα ϕ − ϕ = α−β α−β ∞   α  −1    0 ≤ −LN Hα  · LN Gβ g + LN Hβ ϕ∞ α−β   α  β −1    |ϕ|∞ . + −LN Hα  · LN G0α  · g ∞ + α−β α−β

13.3 Proof of Theorem 1.6

387

By Corollary 13.21, it follows that the first term on the last inequality converges to zero as α → +∞:   α  −1    −LN Hα  · LN G0β g + LN Hβ ϕ∞ −→ 0 as α → +∞. α−β For the second term, by using formula (13.6) with f := 1 and the nonnegativity of G0β and LN G0α we find that       LN G0α  = LN G0α 1 = LN G0β 1 − (α − β)LN G0α G0β 1 ∞ ∞     ≤ LN G0β 1 = LN G0β  for all α > 0. ∞

Hence, by Corollary 13.21 it follows that the second term also converges to zero as α → +∞:   α  −1    −LN Hα  · LN G0α  · g ∞ −→ 0 as α → +∞. α−β It is clear that the third term converges to zero as α → +∞: β |ϕ|∞ −→ 0 α−β

as α → +∞.

Therefore, we have proved assertion (13.37) and hence the desired assertion (13.36). The proof of assertion (13.30) is complete. Step 8: Summing up, we have proved that the operator AN , defined by formula (13.20), satisfies conditions (a) through (d) in Theorem 3.34. Hence it follows from an application of the same theorem that the operator AN is the infinitesimal generator of some Feller semigroup on the state space D. Now the proof of Theorem 13.17 is complete.  

13.3 Proof of Theorem 1.6 We apply part (ii) of Theorem 3.34 (the Hille–Yosida theorem) to the operator A defined by formula (1.6). First, we simplify the boundary condition    ∂u   + γ(x )u = 0 on ∂D. Lu = μ(x ) ∂n ∂D We remark that the original two conditions (A) and (B) are equivalent to the following two conditions (A) and (B ): (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B ) γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}.

388

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Since we have the inequality μ(x ) − γ(x ) > 0

on ∂D,

we find that the boundary condition   ∂u   μ(x ) + γ(x )u = 0 on ∂D ∂n ∂D 

is equivalent to the boundary condition       γ μ(x ) ∂u + u = 0 on ∂D.     μ(x ) − γ(x ) ∂n μ(x ) − γ(x ) ∂D Hence, if we let μ(x ) , μ(x ) − γ(x ) γ(x ) γ 0(x ) := , μ(x ) − γ(x )

μ 0(x ) :=

then we have the assertions

  ∂u   μ 0(x ) +0 γ (x )u = 0 on ∂D ∂n ∂D 

and • 0≤μ 0(x ) ≤ 1 on ∂D, 0(x ) − 1 on ∂D. • γ 0(x ) = μ Namely, we may assume that the boundary condition L is of the form   ∂u Lu = μ(x ) + (μ(x ) − 1)u = 0 on ∂D, ∂n ∂D with

0 ≤ μ(x ) ≤ 1

on ∂D.

Next we express the boundary condition L in terms of the Dirichlet and Neumann conditions. It follows from an application of Lemmas 13.9 and 13.11 that LG0α = μ(x ) LN G0α , and that

LHα = μ(x ) LN Hα + μ(x ) − 1.

Hence, in view of definition (13.18) we obtain that

13.3 Proof of Theorem 1.6

389

Lu

   αI − A u + LHα (u|∂D )    = μ(x )LN G0α αI − A u + μ(x )LN Hα (u|∂D ) + (μ(x ) − 1)(u|∂D )      = μ(x ) LN G0α αI − A u + LN Hα (u|∂D ) + (μ(x ) − 1)(u|∂D )

= LG0α

= μ(x )LN u + (μ(x ) − 1) (u|∂D )

for every u ∈ D(L).

This proves the desired formula L = μ(x )LN + μ(x ) − 1.

(13.38)

Therefore, the next theorem proves Theorem 1.6: Theorem 13.22. We define a linear operator     A : C0 D \ M −→ C0 D \ M as follows (cf. formula (13.15)). (a) The domain D (A) of A is the space      D (A) = u ∈ C0 D \ M : Au ∈ C0 D \ M ,

(13.39) '





Lu = μ(x ) LN u + (μ(x ) − 1) (u|∂D ) = 0 . (b) Au = Au for every u ∈ D (A). Assume that the following condition (A ) is satisfied: (A ) 0 ≤ μ(x ) ≤ 1 on ∂D. Then the operator A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on the state space D \ M , and the Green operator Gα = (αI − A)−1 for α > 0 is given by the following formula:   −1  LGN Gα f = GN (13.40) α f − Hα LHα αf   for every f ∈ C0 D \ M . Here GN α is the Green operator for the Neumann condition LN given by formula    −1 0 LN G0α f (13.21) GN α f = Gα f − Hα LN Hα for every f ∈ C(D), and LGN α is a boundary operator given by the formula   N  for every f ∈ C(D). LGN α f = (μ(x ) − 1) Gα f |∂D

390

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Remark 13.23. (1) We can express the relationships between the operators A, Gα , Hα and LHα in matrix form:     I 0 −1 αI − A Gα Hα LHα = . 0 I L (2) In terms of the Boutet de Monvel calculus, we can express the Green operator Gα in the matrix form (see Figure 13.10 below) ⎛ ⎝

LGN α

⎞

−Hα

GN α

LHα

−1

⎠:

C(D)

C(D) −→

C(∂D)

C(∂D).

LGN

C(D) − −−−α− → C(∂D) GN α

LHα

−1

C(D) ←−−−− − C(∂D) −Hα

Fig. 13.10. The mapping properties of the terms of the Green operator Gα defined by formula (13.40)

Proof. In order to prove Theorem 13.22, we apply part (ii) of Theorem 3.34 (the Hille–Yosida theorem) to the operator A. The proof is divided into six steps. Step 1: First, we prove that: If condition (A ) is satisfied, then the closed operator LHα is the generator of some Feller semigroup on the state space ∂D, for any sufficiently large α > 0. We introduce a linear operator T (α) = LHα : B 1−1/p,p (∂D) −→ B 1−1/p,p (∂D) for N < p < ∞ as follows. (a) The domain D (T (α)) of T (α) is the space   D (T (α)) = ϕ ∈ B 1−1/p,p (∂D) : LHα ϕ ∈ B 1−1/p,p (∂D) . (b) T (α)ϕ = LHα ϕ for every ϕ ∈ D (T (α)).

13.3 Proof of Theorem 1.6

391

Then, by arguing just as in the proof of [121, p. 101, Theorem 7.1] with γ(x ) := μ(x ) − 1, we obtain that The operator T (α) is bijective for any sufficiently large α > 0. Furthermore, it maps the space C ∞ (∂D) onto itself.

(13.41)

Since we have the assertion LHα = T (α)

on C ∞ (∂D),

it follows from assertion (13.41) that the operator LHα : C ∞ (∂D) −→ C ∞ (∂D) is surjective for any sufficiently large α > 0. This implies that the range R(LHα ) is a dense subset of C(∂D). Hence, by applying part (ii) of Theorem 13.14 we obtain that the operator LHα generates a Feller semigroup on the state space ∂D, for any sufficiently large α > 0. The situation can be visualized in Figure 13.11 below. LH

C(∂D) − −−−α −→

C(∂D)

T (α)

D(T (α)) − −−−−→ B 1−1/p,p (∂D)

−−−−→ C ∞ (∂D) − LHα

C ∞ (∂D)

Fig. 13.11. The mapping properties of the operator T (α) and LHα for N < p < ∞

Step 2: Next we prove that: The operator LHβ generates a Feller semigroup, on the state space ∂D for any β > 0. We take a positive constant α so large that the operator LHα generates a Feller semigroup on the state space ∂D. We apply Corollary 3.37 with K := ∂D to the operator LHβ for β > 0. By formula (13.16), it follows that the operator LHβ can be written as

392

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

LHβ = LHα + Mαβ , where Mαβ = (α − β)LG0α Hβ is a bounded linear operator on C(∂D) into itself. Furthermore, assertion (13.15) implies that the operator LHβ satisfies condition (β  ) of Theorem 3.36 (the Hille–Yosida–Ray theorem). Therefore, it follows from an application of Corollary 3.37 that the operator LHβ also generates a Feller semigroup on the state space ∂D. Step 3: Now we prove that: If condition (A ) is satisfied, then the equation

(13.42)

LHα ψ = ϕ

  has a unique solution ψ in D LHα for any ϕ ∈ C(∂D); hence the inverse LHα

−1

of LHα can be defined on the whole space C(∂D).

Furthermore, the operator −LHα on the space C(∂D).

−1

is non-negative and bounded

Since we have, by inequality (13.24),   ∂  LHα 1 = μ(x ) (Hα 1) + (μ(x ) − 1) < 0 ∂n ∂D 

on ∂D,

it follows that kα = min (−LHα 1(x )) = − max LHα 1(x ) > 0.   x ∈∂D

x ∈∂D

In view of Lemma 13.20, we find that the constants kα are increasing in α > 0: α ≥ β > 0 =⇒ kα ≥ kβ . Furthermore, by using Corollary 3.35 with ⎧ ⎪ ⎨K := ∂D, A := LHα , ⎪ ⎩ c := kα , we obtain that the operator LHα + kα I is the infinitesimal generator of some Feller semigroup on the state space ∂D. Therefore, since kα > 0, it follows from an application of part (i) of Theorem 3.34 with A := LHα + kα I that the equation

13.3 Proof of Theorem 1.6

393

  −LHα ψ = kα I − (LHα + kα I) ψ = ϕ   has a unique solution ψ ∈ D LHα for any ϕ ∈ C(∂D), and further that the  −1 −1 operator −LHα = kα I − (LHα + kα I) is non-negative and bounded on the space C(∂D) with norm    −1  1 −1     . −LHα  =  kα I − (LHα + kα I) ≤ kα Step 4: By assertion (13.42), we can define the operator Gα by formula (13.40) for all α > 0. We prove that: Gα = (αI − A)−1

for all α > 0.

(13.43)

13.7 and Theorem 13.17, it follows that we have, for all f ∈  By Lemma C0 D \ M ,   Gα f ∈ D A , and A(Gα f ) = αGα f − f. Furthermore, we obtain that the function Gα f satisfies the boundary condition   −1  = 0 on ∂D. (13.44) L(Gα f ) = LGN LGN α f − LHα LHα αf However, we recall that Lu = μ(x )LN u + (μ(x ) − 1) (u|∂D )

for every u ∈ D(L).

(13.45)

Hence it follows that the boundary condition (13.44) is equivalent to the following: L(Gα f ) = μ(x )LN (Gα f ) + (μ(x ) − 1) (Gα f |∂D ) = 0   In particular, we have, for all f ∈ C0 D \ M ,

on ∂D.

(13.46)

Gα f = 0 on M = {x ∈ ∂D : μ(x ) = 0} , and so A(Gα f ) = αGα f − f = 0 on M . Summing up, we have proved that   f ∈ C0 D \ M =⇒

      Gα f ∈ D (A) = u ∈ C0 D \ M : Au ∈ C0 D \ M , Lu = 0 on ∂D , and further that

394

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

(αI − A)Gα f = f, This proves that (αI − A)Gα = I

  f ∈ C0 D \ M .   on C0 D \ M .

Therefore, in order to prove formula (13.43), it suffices to show the injectivity of the operator αI − A for α > 0. Assume that, for α > 0, u ∈ D (A) and (αI − A)u = 0. Then, by Corollary 13.8 it follows that the function u can be written as follows:   u = Hα (u|∂D ) , u|∂D ∈ D = D LHα . Thus we have the formula LHα (u|∂D ) = Lu = 0 on ∂D. In view of assertion (13.42), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0

in D.

This proves the injectivity of αI − A for α > 0. Step 5: Now we prove the following three assertions (i), (ii) and (iii):   (i) The operator Gα is non-negative on the space C0 D \ M :   f ∈ C0 D \ M , f ≥ 0

on D \ M =⇒ Gα f ≥ 0

on D \ M . (13.47)

  (ii) The operator Gα is bounded on the space C0 D \ M with norm

Gα ≤

1 α

for all α > 0.

(13.48)

  (iii) The domain D (A) is dense in the space C0 D \ M . Step 5-1: In order to prove assertion (i), we have only to show the nonnegativity of the operator Gα on the space C(D): f ∈ C(D), f ≥ 0

on D =⇒ Gα f ≥ 0

Recall that the Dirichlet problem  (α − A)u = f u|∂D = ϕ

in D, on ∂D

on D.

(13.49)

(13.2)

13.3 Proof of Theorem 1.6

is uniquely solvable. Hence it follows that  N  0 GN α f = Hα Gα f |∂D + Gα f

on D.

395

(13.50)

Indeed, the both sides satisfy the same equation (α − A)u = f in D and have  the same boundary values GN α f ∂D on ∂D. Thus, by applying the boundary operator L to the both sides of formula (13.50) we obtain that  N  0 LGN α f = LHα Gα f |∂D + LGα f. Since the operators −LHα f ≥0

−1

and LG0α are non-negative, it follows that

on D

=⇒       −1  −1 N 0f −LHα LGN f = −G f | + −LH LG ∂D α α α α ≥ −GN α f |∂D

on ∂D.

Therefore, by the non-negativity of Hα and G0α we find from formulas (13.40) and (13.50) that   −1  N −LH LG Gα f = GN f + H f α α α α   N N ≥ Gα f + Hα −Gα f |∂D = G0α f ≥0

on D.

This proves the desired assertion (13.49) and hence assertion (i). Step 5-2: Next we prove assertion (ii). To do this, it suffices to show the boundedness of the operator Gα on the space C(D): Gα 1 ≤

1 α

on D,

since Gα is non-negative on the space C(D). We remark (cf. formula (13.45)) that   N  N  LGN α f = μ(x ) LN Gα f + (μ(x ) − 1) Gα f |∂D   = (μ(x ) − 1) GN α f |∂D , so that

  −1  LGN Gα f = GN α f − Hα LHα αf   −1   N −LH (μ(x . f + H ) − 1) G f | = GN α α ∂D α α

Hence, by using this formula with f := 1 we obtain that

(13.51)

396

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

  −1  (1 − μ(x ))GN . Gα 1 = GN α 1 − Hα −LHα α 1|∂D However, we have, by inequality (13.27), 0 ≤ GN α1≤ and also

1 α

on D,

  −1  Hα −LHα (1 − μ(x )) GN ≥0 1| ∂D α

since the operators Hα and −LHα on ∂D. Therefore, we obtain that

−1

on D,

are non-negative and since 1 − μ(x ) ≥ 0

0 ≤ Gα 1 ≤ GN α1 ≤

1 α

on D.

This proves the desired assertion (13.51) and hence assertion (ii). Step 5-3: Finally, we prove assertion (iii). In view of formula (13.43), it suffices to show that lim αGα f − f ∞ = 0 (13.52)   for every f ∈ C0 D \ M ∩ C ∞ (D),     since the space C0 D \ M ∩ C ∞ (D) is dense in C0 D \ M . We remark that   −1  (13.53) αGα f − f = αGN LGN α f − f − αHα LHα αf      N  −1 α(1 − μ(x ))GN . = αGα f − f + Hα LHα α f |∂D α→+∞

We estimate the last two terms of formula (13.53) as follows: (1) By assertion (13.36), it follows that the first term of formula (13.53) tends to zero as α → +∞:    lim αGN (13.54) α f − f ∞ = 0. α→+∞

(2) To estimate the second term of formula (13.53), we remark that   −1  Hα LHα f | α (1 − μ(x )) GN (13.55) ∂D α   −1 = Hα LHα ((1 − μ(x ))f |∂D )     −1  (1 − μ(x )) αGN . f − f | + Hα LHα ∂D α However, it follows that the second term in the right-hand side of formula (13.55) tends to zero as α → +∞. Indeed, we have, by assertion (13.54),

13.3 Proof of Theorem 1.6

     −1    (1 − μ(x )) αGN Hα LHα  α f − f |∂D ∞      N  −1      ≤ −LHα  · (1 − μ(x )) αGα f − f |∂D ∞    1   (1 − μ(x )) αGN ≤ α f − f |∂D ∞ kα  1  αGN f − f  −→ 0 as α → +∞. ≤ α ∞ k1

397

(13.56)

Here we have used the following two inequalities (cf. the proof of assertion (13.42)):   1 −1   • −LHα  ≤ for all α > 0. kα • k1 = min (−LH1 1)(x ) ≤ kα = min (−LHα 1)(x ) for all α ≥ 1.   x ∈∂D

x ∈∂D

Thus we are reduced to the study of the first term of the right-hand side of formula (13.55)   −1 Hα LHα ((1 − μ(x ))f |∂D ) . Now, for any given ε > 0, we can find a function h(x ) in C ∞ (∂D) such that  h = 0 near M = {x ∈ ∂D : μ(x ) = 0},

(1 − μ(x )) f |∂D − h ∞ < ε. Then we have, for all α ≥ 1,      −1 −1   Hα LHα ((1 − μ(x ))f |∂D ) − Hα LHα h  ∞   ε −1    ≤ −LHα  · |(1 − μ(x ))f |∂D − h|∞ ≤ kα ε ≤ . k1

(13.57)

Furthermore, we can find a function θ(x ) in C0∞ (∂D) such that  θ(x ) = 1 near M , on ∂D. (1 − θ(x ))h(x ) = h(x ) Then we have the assertion

  1 − θ(x ) h(x ) = (1 − θ(x )) h(x ) = (−LHα 1(x )) h(x ) −LHα 1(x )    1−θ    ≤  −LHα 1  · |h|∞ (−LHα 1(x )) . ∞

Since the operator −LHα that

−1

is non-negative on the space C(∂D), it follows

398

13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

−LHα

−1

   1−θ    h≤ −LHα 1 

so that      −1 −1     Hα LHα h  ≤ −LHα h

∞

∞

· |h|∞

on ∂D,

   1−θ   · |h| . ≤  ∞ −LHα 1 ∞

(13.58)

However, there exists a constant δ0 > 0 such that 0≤

1 − θ(x ) ≤ δ0 μ(x )

for all x ∈ ∂D.

Thus it follows that 1 − θ(x ) = −LHα 1(x )

1 − θ(x )   ∂   (Hα 1(x )) + (1 − μ(x )) μ(x ) − ∂n   1 − θ(x ) 1   ≤ ∂ μ(x )  (Hα 1(x )) − ∂n 1  for all x ∈ ∂D,  ≤ δ0 ∂  (Hα 1(x )) minx ∈∂D − ∂n

and hence from Lemma 13.20 that    1−θ    = 0. lim α→+∞  −LHα 1  ∞

(13.59)

Summing up, we obtain from inequalities (13.57) and (13.58) and assertion (13.59) that    −1   lim sup Hα LHα ((1 − μ(x ))f |∂D )  ∞ α→+∞    −1   ≤ lim sup Hα LHα h  ∞ α→+∞       −1 −1   + Hα LHα ((1 − μ(x ))f |∂D ) − Hα LHα h  ∞    1−θ   · |h| + ε ≤ lim  ∞ α→+∞ −LHα 1  k1 ∞ ε = . k1 Since ε is arbitrary, this proves that the first term of the right-hand side of formula (13.55) tends to zero as α → +∞:

13.5 Notes and Comments

   −1   lim Hα LHα ((1 − μ(x ))f |∂D ) 

α→+∞

∞

= 0.

399

(13.60)

By assertions (13.56) and (13.60), we obtain that the last term of formula (13.53) also tends to zero:    −1    α(1 − μ(x ))GN lim Hα LHα f | (13.61)  = 0. ∂D α α→+∞

∞

Therefore, the desired assertion (13.52) follows by combining assertions (13.54) and (13.61). The proof of assertion (iii) is complete. Step 6: Summing up, we have proved that the operator A, defined by formula (13.39), satisfies conditions (a) through (d) in Theorem 3.34. Hence, in view of assertion (13.36), it follows from an application of part (ii) of the same theorem that the operator A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on the state space D \ M . Now the proof of Theorem 13.22 and hence that of Theorem 1.6 is complete.  

13.4 Proof of Part (ii) of Theorem 1.5 We apply Theorem 2.2 to the operator A. In the proof of Theorem 13.22,   we have proved that the domain D (A) is dense in the space C0 D \ M . Furthermore, part (i) of Theorem 1.5 verifies conditions (2.1) and (2.2). Therefore, it follows from an application of Theorem 2.2 that: The semigroup Tt can be extended to a semigroup Tz that is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2. This (together with Theorem 1.6) proves part (ii) of Theorem 1.5.  

13.5 Notes and Comments This chapter is an expanded and revised version of Chapter 9 of the second edition [121], which is adapted from Bony–Courr`ege–Priouret [15], Sato–Ueno [98] and Taira [114], [116] and [122].

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

In this chapter we prove Theorems 1.8, 1.9, 1.10 and 1.11, generalizing Theorems 1.4, 1.5 and 1.6 for second-order, elliptic Waldenfels operators. More precisely, we consider a second-order, elliptic Waldenfels operator W with real coefficients such that W u(x) = Au(x) + Su(x) (1.2) ⎛ ⎞ N N   ∂2u ∂u := ⎝ aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ ∂x ∂x ∂x i j i i,j=1 i=1 ⎛ ⎞  N  ∂u ⎝u(x + z) − u(x) − zj (x)⎠ s(x, z) m(dz). + ∂x N j R \{0} j=1 Here: (1) aij ∈ C ∞ (D), aij (x) = aji (x) for all 1 ≤ i, j ≤ N , and there exists a constant a0 > 0 such that N 

aij (x)ξi ξj ≥ a0 |ξ|2

for all x ∈ D and ξ ∈ RN .

i,j=1 ∞

(2) b ∈ C (D) for all 1 ≤ i ≤ N . (3) c ∈ C ∞ (D), and c(x) ≤ 0 in D, but c(x) ≡ 0 in D. (4) s(x, z) ∈ L∞ (RN × RN ) and 0 ≤ s(x, z) ≤ 1 almost everywhere in the product space RN × RN , and there exist constants C0 > 0 and 0 < θ0 < 1 such that i

|s(x, z) − s(y, z)| ≤ C0 |x − y|

θ0

(1.3a)

for all x, y ∈ D and almost all z ∈ R , N

and the support condition

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 14

402

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

s(x, z) = 0

if x ∈ D and x + z ∈ D.

(1.3b)

Probabilistically, the support condition (1.3b) implies that all jumps from D are within D. Analytically, the support condition (1.3b) guarantees that the integral operator S may be considered as an operator acting on functions u defined on the closure D (see [48, Chapter II, Remark 1.19]). (5) The measure m(dz) is a Radon measure on RN \ {0} which has a density with respect to the Lebesgue measure dz on RN , and satisfies the moment condition (see Example 1.1)   2 |z| m(dz) + |z| m(dz) < ∞. (1.4) {01}

(6) Finally, we assume that W 1(x) = A1(x) + S1(x) = c(x) ≤ 0 and c(x) ≡ 0 in D.

(1.5)

In Section 14.1, by using the H¨older space theory of pseudo-differential operators we study the non-homogeneous boundary value problem ⎧ ⎨Au = f in D, (∗) ∂u   ⎩Lu = μ(x ) + γ(x )u = ϕ on ∂D, ∂n in the framework of H¨ older spaces, and prove an existence and uniqueness theorem for the problem (∗) (Theorem 14.1). More precisely, we prove that if the two conditions (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D, (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D, are satisfied, then the mapping (A, L) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is an algebraic and topological isomorphism for all 0 < θ < 1. The function space C1+θ (∂D) is introduced in Subsection 1.2.2 under conditions (A) and (B). In Section 14.2, we prove an existence and uniqueness theorem for the non-homogeneous boundary value problem (Theorem 1.8)  W u = f in D, (∗∗) Lu = ϕ on ∂D. in the framework of H¨ older spaces. Namely, we prove that if conditions (A) and (B) and the condition (H) The integral operator S satisfies conditions (1.3a), (1.3b), (1.4) and (1.5),

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

403

are satisfied, then the mapping (W, L) = (A + S, L) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is an algebraic and topological isomorphism for all 0 < θ ≤ θ0 . The proof of Theorem 1.8 is flowcharted (Table 14.1). Due to the non-local character of the Waldenfels integro-differential operator W = A + S, we find more difficulties in the bounded domain D than in the whole space RN . In fact, when considering the Dirichlet problem in D, it is natural to use the zero extension of functions in the interior D outside of the closure D = D ∪ ∂D. This extension has a probabilistic interpretation. Namely, this corresponds to stopping the diffusion process with jumps in the whole space RN at the first exit time of the closure D. However, the zero extension produces a singularity of solutions at the boundary ∂D. In order to remove this singularity, we introduce various conditions on the structure of jumps for the Waldenfels integro-differential operator W = A + S such as conditions (1.3a), (1.3b) and (1.4). More precisely, we can estimate the L´evy integro-differential operator S in terms of H¨ older norms, just as in [48, Chapter II, Lemmas 1.2 and 1.5] (Lemmas 14.4 and 14.5), and prove that the L´evy operator S : C 2+θ (D) −→ C θ (D) is compact for all 0 < θ ≤ θ0 under conditions (1.3a), (1.3b) and (1.4) (Lemma 14.6). This implies that the mapping (W, L) = (A, L) + (S, 0) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is a perturbation of a compact operator to the mapping (A, L) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D). In this way, the proof of Theorem 1.8 can be reduced to the differential operator case (Theorem 14.1). Hence we have, by Theorem 14.1 and Lemma 14.6, ind (W, L) = ind (A, L) = 0, since the index is stable under compact perturbations ([51]). On the other hand, by using the maximum principle (Theorem 10.7 and Lemma 10.11) we find that if conditions (A), (B) and (1.5) are satisfied, then the mapping (W, L) is injective (Proposition 14.7). Therefore, we can prove that if conditions (A), (B) and (H) are satisfied, then the mapping (W, L) is bijective (Theorem 1.8). In Section 14.3 we prove Theorem 1.9 (Theorem 14.8). We estimate the integral operator S in terms of Lp norms, and show that S is an Ap -completely continuous operator in the sense of Gohberg–Kre˘ın [51] (Lemmas 14.9 and 14.10). The proof of Theorem 1.9 is flowcharted (Table 14.2).

404

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

Section 14.4 is devoted to the proof of Theorem 1.10. Theorem 1.10 follows from Theorem 1.9 by using Sobolev’s imbedding theorems and a λ-dependent localization argument. The proof is carried out in a chain of auxiliary lemmas (Lemmas 14.11, 14.12 and 14.15). The proof of Theorem 1.10 is flowcharted (Table 14.3). In Section 14.5, as an application, we construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it “dies” at the time when it reaches the set where the particle is definitely absorbed, generalizing Theorem 1.6 (Theorem 1.11). The proof of Theorem 1.11 is flowcharted (Table 14.4).

14.1 Existence and Uniqueness Theorem in H¨ older Spaces In this section, by using the H¨older space theory of pseudo-differential operators we study the boundary value problem (∗) in the framework of H¨ older spaces. First, we prove an existence and uniqueness theorem for the nonhomogeneous boundary value problem (∗) ([117, Theorem 1.1]): Theorem 14.1 (the existence and uniqueness theorem). Let 0 < θ < 1. Assume that the following two conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D. Then the mapping (A, L) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is an algebraic and topological isomorphism. In particular, for any f ∈ C θ (D) and any ϕ ∈ C1+θ (∂D), there exists a unique solution u ∈ C 2+θ (D) of problem (∗). Proof. In order to prove Theorem 14.1, it suffices to show that the mapping (A, L) is bijective. Indeed, the continuity of the inverse of (A, L) follows immediately from an application of Banach’s open mapping theorem (see [20, Chapter 2, Corollary 2.7], [44, Chapter 4, Theorem 4.6.2], [147, Chapter II, Section 5, Corollary]), since (A, L) is a continuous operator. The proof is divided into four steps. Step 1: Let (f, ϕ) be an arbitrary element of the space C θ (D) ⊕ C1+θ (∂D) with ϕ = μ(x )ϕ1 − γ(x )ϕ2

for ϕ1 ∈ C 1+θ (∂D) and ϕ2 ∈ C 2+θ (∂D).

14.1 Existence and Uniqueness Theorem in H¨ older Spaces

405

First, we show that the boundary value problem (∗) can be reduced to the study of an operator on the boundary, just as in Section 6.3. To do this, we consider the following Neumann problem: ⎧ ⎨Av = f in D, (14.1) ∂v ⎩ = ϕ1 on ∂D. ∂n Recall that the existence and uniqueness theorem for problem (14.1) is well established in the framework of H¨older spaces (see Gilbarg–Trudinger [50, Theorem 6.31]). Thus we find that a function u ∈ C 2+θ (D) is a solution of problem (∗) if and only if the function w = u − v ∈ C 2+θ (D) is a solution of the problem  Aw = 0 Lw = ϕ − Lv = μ(x )ϕ1 − γ(x )ϕ2 − Lv

in D, on ∂D.

Here it should be noticed that Lv = μ(x ) so that

∂v + γ(x ) (v|∂D ) = μ(x )ϕ1 + γ(x ) (v|∂D ) , ∂n

Lw = −γ(x ) (ϕ2 + (v|∂D )) ∈ C 2+θ (∂D).

However, we know that every solution w ∈ C 2+θ (D) of the homogeneous equation: Aw = 0 in D can be expressed as follows (see [50, Theorem 6.14]): w = Pψ

for ψ ∈ C 2+θ (∂D).

Thus we can reduce the study of problem (∗) to that of the pseudo-differential equation T ψ := L (P ψ) = Lw = −γ(x ) (ϕ2 + (v|∂D ))

on ∂D.

(14.2)

We remark that the pseudo-differential equation (14.2) is a generalization of the classical Fredholm integral equation. Summing up, we have proved the following proposition: Proposition 14.2. For given functions f ∈ C θ (D) and ϕ ∈ C1+θ (∂D), there exists a solution u ∈ C 2+θ (D) of problem (∗) if and only if there exists a solution ψ ∈ C 2+θ (∂D) of the pseudo-differential equation (14.2). Step 2: We study the operator T = LP in question. The next proposition is an essential step in the proof of Theorem 14.1 (see Lemma 7.1):

406

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

Proposition 14.3. If conditions (A) and (B) are satisfied, then there exists a parametrix E in the H¨ ormander class L01,1/2 (∂D) for T which maps the H¨ older space C k+θ (∂D) continuously into itself, for any non-negative integer k. Proof. Indeed, by virtue of [121, p. 91, Lemma 5.2] (cf. [68, Theorem 3.1]) we can construct a parametrix E in the H¨ ormander class L01,1/2 (∂D) for T . Furthermore, it follows from an application of the Besov space boundedness theorem (Theorem 4.47) that the parametrix E maps the H¨ older space C k+θ (∂D) continuously into itself, for any nonnegative integer k, since we have the formula (see [135, Theorem 2.5.7]) k+θ C k+θ (∂D) = B∞,∞ (∂D).

The proof of proposition 14.3 is complete.   Step 3: Now we remark that  C θ (D) ⊂ Lp (D), 1−1/p,p C1+θ (∂D) ⊂ B (∂D) for 1 < p < ∞. Thus, by applying [123, Theorem 1.1] to our situation we find that problem (∗) has a unique solution u ∈ H 2,p (D) for any f ∈ C θ (D) and any ϕ ∈ C1+θ (∂D). Furthermore, by virtue of Proposition 14.3 it follows that the solution u can be written in the form u = v + Pψ

for v ∈ C 2+θ (D) and ψ ∈ B 2−1/p,p (∂D).

(14.3)

However, we have, by equation (14.2) and Proposition 14.3, ψ ∈ C 2+θ (∂D). Indeed, it suffices to note that ψ ≡ E(T ψ) = −E (γ(x )(ϕ2 + (v|∂D )) ∈ C 2+θ (∂D) mod C ∞ (∂D). Therefore, we obtain from formula (14.3) that u = v + P ψ ∈ C 2+θ (D). Step 4: The injectivity of (A, L) follows from an application of the maximum principle (Theorem 10.7 and Lemma 10.11 with S ≡ 0). The proof of Theorem 14.1 is now complete.  

14.2 Proof of Theorem 1.8 In this section we study problem (∗∗) in the framework of H¨older spaces, and prove Theorem 1.8 under conditions (A), (B) and (H). The essential point

14.2 Proof of Theorem 1.8

407

Proposition 14.2 (reduction to ∂D) Theorem 14.1 (ind (A, L) = 0) Proposition 14.3 (parametrix E for T = LP ) Theorem 1.8 (bijectivity of (W, L)) Lemma 14.6 (compact perturbation) ind (W, L) = 0 (injectivity of (W, L)) Proposition 14.7 (maximum principle)

Table 14.1. A flowchart for the proof of Theorem 1.8

in the proof is to estimate the integral operator S in terms of H¨ older norms. We show that the operator (W, L) may be considered as a perturbation of a compact operator to the operator (A, L) in the framework of H¨older spaces. Thus the proof of Theorem 1.8 is reduced to the differential operator case. The proof of Theorem 1.8 can be flowcharted as in Table 14.1 above. The proof of Theorem 1.8 is divided into three steps. Step (I): By virtue of Claim 10.8, we can estimate the term Su in terms of H¨ older norms, just as in [48, Chapter II, Lemmas 1.2 and 1.5]: Lemma 14.4. Assume that condition (H) is satisfied. Then, for every η > 0 there exists a constant Cη > 0 such that we have, for all u ∈ C 2 (D),  

Su ∞ ≤ η ∇2 u∞ + Cη ( u ∞ + ∇u ∞ ) . (14.4) Here

u ∞ = sup |u(x)| = max |u(x)| . x∈D

x∈D

Proof. By using Taylor’s formula, we decompose the integral term Su into the following two terms: Su(x)  =

⎛

RN \{0}

=

⎝u(x + z) − u(x) −

N   i,j=1

{0ε} j=1  1  = (1 − t) dt z · ∇2 u(x + tz)z s(x, z) m(dz) 0 {0ε}

:= Sε(1) u(x) + Sε(2) u(x). (1)

(2)

By Claim 10.8, we can estimate the terms Sε u and Sε u as follows:    (1)  Sε u $ ∞ %     1 1 2 ≤ |z| m(dz) ∇2 u∞ = σ(ε) ∇2 u∞ , 2 2 {0ε}

% m(dz) u ∞ +

$

%

{|z|>ε}

|z| m(dz) ∇u ∞

= 2τ (ε) u ∞ + δ(ε) ∇u ∞     C1 C1 + C2 ∇u ∞ . ≤2 + C2 u ∞ + ε2 ε Therefore, we have proved that        

Su ∞ ≤ Sε(1) u + Sε(2) u ∞ ∞     2  C1 C1 1   + C2 ∇u ∞ . ≤ σ(ε) ∇ u ∞ + 2 + C2 u ∞ + 2 ε2 ε In view of assertion (10.12a), this proves the desired estimate (14.4) if we choose ε sufficiently small. The proof of Lemma 14.4 is complete.   Lemma 14.5. Assume that condition (H) is satisfied. Then, for every η > 0 there exists a constant Cη > 0 such that we have, for all u ∈ C 2+θ0 (D),     . (14.6)

Su θ0 ≤ η ∇2 u θ + Cη u θ0 + ∇u θ0 C

(D)

C

0 (D)

C

(D)

Here

u C θ0 (D) = u ∞ + [u]θ0 ,

[u]θ0 = sup

x,y∈D x =y

C

(D)

|u(x) − u(y)| . |x − y|θ0

14.2 Proof of Theorem 1.8 (1)

409

(2)

Proof. We estimate the terms Sε u and Sε u of formula (14.5) in terms of H¨older norms. (2) (2) In order to estimate the term Sε u, we write the difference Sε u(x) − (2) Sε u(y) in the following form: Sε(2) u(x) − Sε(2) u(y)  = (u(x + z) − u(y + z)) s(x, z) m(dz) {|z|>ε}  u(y + z) (s(x, z) − s(y, z)) m(dz) + {|z|>ε}  + (u(y) − u(x)) s(x, z) m(dz) {|z|>ε}  − u(y) (s(x, z) − s(y, z)) m(dz) {|z|>ε}  (∇u(x) − ∇u(y)) · z s(x, z) m(dz) − {|z|>ε}  − ∇u(y) · z (s(x, z) − s(y, z)) m(dz) {|z|>ε}

:= A(x, y) + B(x, y) + C(x, y) − D(x, y) − E(x, y) − F (x, y). Then, by using estimates (10.12c), (10.12b) and condition (1.3a) we can estimate the terms A(x, y) through F (x, y) as follows: • |A(x, y| , |C(x, y)| ≤ [u]θ0 τ (ε) |x − y|θ0   C1 ≤ + C [u]θ0 |x − y|θ0 . 2 ε2 • |E(x, y)| ≤ [∇u]θ0 δ(ε) |x − y|θ0   C1 + C2 [∇u]θ0 |x − y|θ0 . ≤ ε θ0

• |B(x, y)| , |D(x, y)| ≤ u ∞ C0 τ (ε) |x − y|   C1 ≤ C0 + C2 u ∞ |x − y|θ0 . ε2 • |F (x, y)| ≤ ∇u ∞ C0 δ(ε) |x − y|θ0   C1 θ + C2 ∇u ∞ |x − y| 0 . ≤ C0 ε

Summing up, we have proved that     D C C1 C1 (2) ≤2 + C2 [u]θ0 + 2C0 + C2 u ∞ Sε u ε2 ε2 θ0

(14.7)

410

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11



   C1 C1 + C2 [∇u]θ0 + C0 + C2 ∇u ∞ ε ε   C1 ≤ 2(1 + C0 ) + C2 u C θ0 (D) ε2   C1 + C2 ∇u C θ0 (D) . + (1 + C0 ) ε +

(1)

The term Sε u can be estimated in a similar way: C

Sε(1) u

D θ0

  1 1 σ(ε) ∇2 u θ0 + σ(ε) C0 ∇2 u ∞ 2 2   1 + C0 ≤ σ(ε) ∇2 u C θ0 (D) . 2

≤

Thus, by combining estimates (14.7) and (14.8) we obtain that C D C D [Su]θ0 ≤ Sε(1) u + Sε(2) u θ0 θ0    2    ≤ η ∇ u C θ0 (D) + Cη u C θ0 (D) + ∇u C θ0 (D) ,

(14.8)

(14.9)

if we choose ε sufficiently small. Therefore, the desired estimate (14.6) follows by combining estimates (14.4) and (14.9). The proof of Lemma 14.5 is complete.   Step (II): We find from Theorem 14.1 that ind (A, L) = 0. However, the next lemma asserts that the L´evy integro-differential operator S : C 2+θ (D) −→ C θ (D) is compact for all 0 < θ ≤ θ0 ; hence that the mapping (W, L) = (A, L) + (S, 0) : C 2+θ (D) −→ C θ (D) ⊕ C1+θ (∂D) is a perturbation of a compact operator to the mapping (A, L) under conditions (1.3a), (1.3b) and (1.4). Lemma 14.6. If condition (H) is satisfied, then the L´evy integro-differential operator S : C 2+θ (D) −→ C θ (D) is compact for all 0 < θ ≤ θ0 .

14.2 Proof of Theorem 1.8

411

Proof. We consider the following two cases (1) and (2). (1) The case where θ = θ0 : Let {uj } be an arbitrary bounded sequence in C 2+θ0 (D); hence there exists a constant K > 0 such that

uj C 2+θ0 (D) ≤ K. Then it follows from an application of the Ascoli–Arzel`a theorem that the injection C 2+θ0 (D) −→ C 1+θ0 (D) is compact (see [50, Lemma 6.36]). Hence we may assume that the sequence {uj } itself is a Cauchy sequence in C 1+θ0 (D). Then, by applying estimate (14.6) to the sequence {uj − uk } we obtain that  

Suj − Suk C θ0 (D) ≤ η ∇2 uj − ∇2 uk C θ0 (D)   + Cη uj − uk C θ0 (D) + ∇uj − ∇uk C θ0 (D) ≤ 2ηK + Cη uj − uk C 1+θ0 (D) . Hence we have the inequality lim sup Suj − Suk C θ0 (D) ≤ 2ηK. j,k→∞

This proves that the sequence {Suj } is a Cauchy sequence in the space C θ0 (D), since η is arbitrary. (2) The case where 0 < θ < θ0 : We remark that θ0

|x − y|

= |x − y|

θ0 −θ

θ

|x − y| ≤ diam D · |x − y|

θ

for all x, y ∈ D,

where diam D is the diameter of the domain D defined by the formula diam D = max |x − y| . x,y∈D

Hence, just as in the proof of estimate (14.9) we can prove that C D C D [Su]θ ≤ Sε(1) u + Sε(2) u θ θ    2    ≤ η ∇ u C θ (D) + Cη u C θ (D) + ∇u C θ (D) . Indeed, it suffices to note that, in the proof of Lemma 14.5 we have the following four estimates: • |A(x, y| , |C(x, y)| ≤ τ (ε) [u]θ |x − y|θ . • |E(x, y)| ≤ δ(ε) [∇u]θ |x − y|θ . θ0

• |B(x, y)| , |D(x, y)| ≤ u ∞ C0 τ (ε) |x − y|

412

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11 θ

≤ τ (ε) C0 u ∞ diam D |x − y| . • |F (x, y)| ≤ ∇u ∞ C0 δ(ε) |x − y|

θ0

≤ δ(ε) C0 diam D ∇u ∞ |x − y|θ . In this way, we can obtain the following estimate for 0 < θ < θ0 , similar to estimate (14.6):

Su C θ (D)     ≤ η ∇2 uC θ (D) + Cη u C θ (D) + ∇u C θ (D)

for all u ∈ C 2+θ (D).

Therefore, just as in the case (1) we find that the operator S : C 2+θ (D) −→ C θ (D) is compact for all 0 < θ < θ0 . The proof of Lemma 14.6 is complete.   Therefore, by combining Theorem 14.1 and Lemma 14.6 we obtain from Gohberg–Kre˘ın [51] that ind (W, L) = ind (A, L) = 0. Step (III): Therefore, in order to show the bijectivity of (W, L) it suffices to prove its injectivity:  u ∈ C 2+θ (D), W u = 0 in D, Lu = 0 on ∂D =⇒ u = 0 in D. However, this is an immediate consequence of the following maximum principle: Proposition 14.7 (the maximum principle). If conditions (A), (B) and (1.5) are satisfied, then we have the assertion  u ∈ C 2 (D), W u ≥ 0 in D, Lu ≥ 0 on ∂D =⇒ u ≤ 0 on D. Proof. (1) If u(x) is a constant m, then we have the assertion 0 ≤ W u(x) = m c(x)

in D.

However, it follows from condition (1.5) that u(x) ≡ m is non-positive, since c(x) ≤ 0 and c(x) ≡ 0 in D. (2) Now we consider the case where u(x) is not a constant. Our proof is based on a reduction to absurdity. Assume, to the contrary, that

14.3 Proof of Theorem 1.9

413

m = max u > 0. D

Then, by applying the strong maximum principle (see Theorem 10.7) to the operator W we obtain that there exists a point x0 of ∂D such that  u(x0 ) = m, u(x) < u(x0 ) for all x ∈ D. Furthermore, it follows from an application of the Hopf boundary point lemma (see Lemma 10.11) that ∂u  (x ) < 0. ∂n 0 Therefore, since we have, by condition (A), 0 ≤ Lu(x0 ) = μ(x0 ) it follows that

∂u  (x ) + γ(x0 ) u(x0 ), ∂n 0

μ(x0 ) = 0 and γ(x0 ) = 0.

This contradicts condition (B). The proof of Proposition 14.7 is complete.   Now the proof of Theorem 1.8 is complete.  

14.3 Proof of Theorem 1.9 In this section we prove Theorem 1.9 under conditions (A), (B) and (H). We estimate the integral operator S in terms of Lp norms, and show that S is an Ap -completely continuous operator in the sense of Gohberg–Kre˘ın [51]. The proof of Theorem 1.9 can be flowcharted as in Table 14.2 below. The next theorem proves Theorem 1.9: Theorem 14.8. Let 1 < p < ∞. Assume that the following three conditions (A), (B) and (H) are satisfied: (A) μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D. (B) μ(x ) − γ(x ) = μ(x ) + |γ(x )| > 0 on ∂D. (H) The integral operator S satisfies conditions (1.3a), (1.3b), (1.4) and (1.5). Then, for every 0 < ε < π/2, there exists a constant rp (ε) > 0 such that the resolvent set of Wp contains the set   Σp (ε) = λ = r2 ei ϑ : r ≥ rp (ε), −π + ε ≤ ϑ ≤ π − ε , and that the resolvent (Wp − λI)−1 satisfies estimate (1.14):   (Wp − λI)−1  ≤ cp (ε) |λ|

for all λ ∈ Σp (ε).

414

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11 Stability of indices under compact perturbations Lemma 14.10 (Ap -complete continuity) Lemma 14.9 (Lp -estimates for S) Theorem 1.9 Theorem 14.8 Estimate (1.9) (resolvent estimate for Ap − λI) Estimate (14.10) (resolvent estimate for Wp − λI) Lemma 14.9 (Lp -estimates for S)

Table 14.2. A flowchart for the proof of Theorem 1.9

Proof. The proof is divided into three steps. Step (i): We show that there exist constants rp (ε) and cp (ε) such that we have, for all λ = r2 ei ϑ satisfying r ≥ rp (ε) and −π + ε ≤ ϑ ≤ π + ε, |u|2,p + |λ|1/2 |u|1,p + |λ| u p ≤ cp (ε) (Wp − λI) u p .

(14.10)

Here

u p = u Lp (D) ,

|u|1,p = ∇u Lp (D) ,

  |u|2,p = ∇2 uLp (D) .

However, we know from estimate (12.4) (see [121, p. 102, estimate (7.1)]) that the desired estimate (14.10) is proved for the differential operator A: |u|2,p + |λ|1/2 |u|1,p + |λ| u p ≤ cp (ε) (Ap − λI) u p

(12.4)

for all u ∈ D (Ap ). In order to replace the last term

(Ap − λI) u p by the term

(Wp − λI) u p , we need the following Lp -estimate (14.11) for the integral operator S: Lemma 14.9. Assume that condition (H) is satisfied. Then, for every η > 0 there exists a constant Cη > 0 such that we have, for all u ∈ H 2,p (D),   (14.11)

Su p ≤ η |u|2,p + Cη u p + |u|1,p .

14.3 Proof of Theorem 1.9

415

Proof. By using Taylor’s formula,, we decompose the integral term Su into the following three terms: Su(x) := S1 u(x) + S2 u(x) − S3 u(x)   1 (1 − t) dt z · ∇2 u(x + tz)z s(x, z) m(dz) = 0 {0ε}  − z · ∇u(x) s(x, z) m(dz). {|z|>ε}

First, we estimate the Lp norm of the term S3 u. By using estimate (10.12b), we obtain that       C1   + C2 |∇u(x)|. z · ∇u(x) s(x, z) m(dz) ≤ δ(ε) |∇u(x)| ≤    {|z|>ε} ε Hence we have the Lp estimate of the term S3 u:   C1 + C2 ∇u p .

S3 u p ≤ ε Secondly, we have the inequality       C1   u(·) s(·, z) m(dz) ≤ + C2 u p .   {|z|>ε}  ε2 p

Furthermore, by using H¨older’s inequality and Fubini’s theorem we obtain from the support condition (1.3b) that  p      u(x + z) s(x, z) m(dz) dx    N R {|z|>ε} %p $   ≤ |u(x + z)| s(x, z) m(dz) dx RN

{|z|>ε}

 ≤

% $

$

|u(x + z)| s(x, z) m(dz) p

RN

{|z|>ε}



p

m(dz) {|z|>ε}



p/q

= τ (ε)

RN



{|z|>ε}

|u(x + z)|p s(x, z)p m(dz) dx



p/q

 |u(x + z)| s(x, z) dx m(dz) p

= τ (ε)

{|z|>ε}



RN

 $

p

≤ τ (ε)

|u(y)| dy

p/q D

%p/q

p

% m(dz)

{|z|>ε}

dx

416

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11 p

= τ (ε)p u p . By estimate (10.12c), we have the Lp estimate of the term S2 u:   C1 + C

S2 u p ≤ 2 u p . ε2 Similarly, by using H¨ older’s inequality and Fubini’s theorem we find that  p    1    2 z · ∇ u(x + tz)z s(x, z) m(dz) dx  (1 − t) dt  RN  0 {0 0 there exists a constant r(ε) > 0 such that if λ = r2 ei ϑ with r ≥ r(ε) and −π + ε ≤ ϑ ≤ π − ε, we have the inequality |λ|

1/2

u C 1 (D) + |λ| u C(D) ≤ c(ε) (W − λI) u C(D) for all u ∈ D (W),

with a constant c(ε) > 0. Proof. (1) First, we show that the domain D (W)

(14.19)

14.4 Proof of Theorem 1.10

421

      = u ∈ C0 D \ M ∩ H 2,p (D) : W u ∈ C0 D \ M , Lu = 0 on ∂D is independent of N < p < ∞. We let Dp       = u ∈ H 2,p (D) ∩ C0 D \ M : W u ∈ C0 D \ M , Lu = 0 on ∂D . Since we have Lp1 (D) ⊂ Lp2 (D) for p1 > p2 , it follows that Dp1 ⊂ Dp2

if p1 > p2 .

Conversely, let v be an arbitrary element of Dp2 :     v ∈ H 2,p2 (D) ∩ C0 D \ M , W v ∈ C0 D \ M , Lv = 0.   Then, since we have v, W v ∈ C0 D \ M ⊂ Lp1 (D), it follows from an application of Theorem 14.8 with p := p1 that there exists a unique function u ∈ H 2,p1 (D) such that  (W − λ)u = (W − λ)v in D, Lu = 0 on ∂D, if we choose λ sufficiently large. Hence we have u − v ∈ H 2,p2 (D) and  (W − λ)(u − v) = 0 in D, L(u − v) = 0 on ∂D. Therefore, by applying again Theorem 14.8 with p := p2 we obtain that u − v = 0 in D, so that v = u ∈ H 2,p1 (D). This proves that v ∈ Dp1 . (2) We shall make use of a λ-dependent localization argument in order to adjust the term (W −λ)u p in inequality (14.15) to obtain inequality (14.19), just as in Section 12.2. (2-a) If x0 is a point of ∂D and if χ is a smooth coordinate transformation such that χ maps B(x0 , η0 ) ∩ D into B(0, δ) ∩ RN + and flattens a part of the boundary ∂D into the plane xN = 0 (see Figure 12.2), then we let • G0 = B(x0 , η0 ) ∩ D, • G = B(x0 , η) ∩ D for 0 < η < η0 , • G = B(x0 , η/2) ∩ D

for 0 < η < η0 .

Similarly, if x0 is a point of D, then we let

422

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

• G0 = B(x0 , η0 ) ⊂ D, • G = B(x0 , η) for 0 < η < η0 , • G = B(x0 , η/2) for 0 < η < η0 . (2-b) We take a function θ(t) ∈ C0∞ (R) such that θ equals 1 near the origin, and define a localizing function      |x − x0 |2 xN − t (14.20) θ ϕ0 (x, η) := θ for x0 = (x0 , t). η2 η Then we have the following claim, analogous to Claim 12.3: Claim 14.13. If u ∈ D (W), then it follows that ϕ0 u ∈ D (Wp ). Proof. First, we recall that u ∈ H 2,p (D)

for all 1 < p < ∞.

Hence we have the assertion ϕ0 u ∈ H 2,p (D). Furthermore, it is easy to verify that the function ϕ0 u satisfies the boundary condition L(ϕ0 u) = 0 on ∂D. Summing up, we have proved that ϕ0 u ∈ D (Wp )

for all 1 < p < ∞.

The proof of Claim 14.13 is complete.   (3) Now let u be an arbitrary element of D (W). Then, by Claim 14.13 we can apply inequality (14.15) to the function ϕ0 u to obtain that 1/2

u C 1 (G ) + |λ| u C(G )

1/2

ϕ0 u C 1 (G ) + |λ| ϕ0 u C(G )

|λ|

≤ |λ|

(14.21)

= |λ|1/2 ϕ0 u C 1 (D) + |λ| ϕ0 u C(D) ≤ Cp (ε) |λ|

N/2p

(W − λ)(ϕ0 u) Lp (D)

for all u ∈ D (W).

(3-a) We estimate the last term (W − λ)(ϕ0 u) Lp(D) in terms of the supremum norm of C(D). First, we write the term (W − λ)(ϕ0 u) in the following form: (W − λ)(ϕ0 u) = ϕ0 ((W − λ)u) + [A, ϕ0 ] u + [S, ϕ0 ] u,

14.4 Proof of Theorem 1.10

423

where [A, ϕ0 ] and [S, ϕ0 ] are the commutators of A and ϕ0 and of S and ϕ0 , respectively: • [A, ϕ0 ] u = A(ϕ0 u) − ϕ0 Au, • [S, ϕ0 ] u = S(ϕ0 u) − ϕ0 Su. Since we have, for some constant c > 0, |G | ≤ |B(x0 , η)| ≤ cη N , it follows from an application of Claim 12.4 that

ϕ0 (W − λ)u Lp (D) = ϕ0 (W − λ)u Lp (G )

(14.22)

≤ c1/p η N/p (W − λ)u C(G ) ≤ c1/p η N/p (W − λ)u C(D)

for all u ∈ D (W).

On the other hand, we can estimate the commutators [A, ϕ0 ] u and [S, ϕ0 ] u as follows: Claim 14.14. We have, as η ↓ 0, • [A, ϕ0 ] u Lp (D)   ≤ C η −1+N/p u C 1 (D) + η −2+N/p u C(D) ,

(14.23a)

• [S, ϕ0 ] u Lp (D)   ≤ C η −1+N/p u C 1 (D) + η −2+N/p u C(D) .

(14.23b)

Proof. By formula (14.20), we have, as η ↓ 0,   |Dα ϕ0 | = O η −|α| . Hence it follows from an application of Claim 12.4 that    ∂ϕ0 ∂u  1 −1+N/p   |u|C 1 (G ) ,  ∂xi ∂xj  p  ≤ C η |u|1,p,G ≤ Cη L (G )  2   ∂ ϕ0  1 −2+N/p   u |u|C(G ) ,  ∂xi ∂xj  p  ≤ C η 2 |u|Lp (G ) ≤ Cη L (G )    ∂ϕ0  1 −1+N/p   |u|C(G ) .  ∂xi u p  ≤ C η |u|Lp (G ) ≤ Cη L (G ) Therefore, we obtain that

[A, ϕ0 ] u Lp (G ) ≤ Cη −1+N/p |u|C 1 (G ) + η −2+N/p |u|C(G ) ≤ Cη −1+N/p |u|C 1 (D) + η −2+N/p |u|C(D) .

424

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

This proves estimate (14.23a). In order to prove estimate (14.23b), we remark that S(ϕ0 u)(x)   = ϕ0 (x + z)u(x + z) − ϕ0 (x)u(x) RN \{0}

 − z · ∇(ϕ0 u)(x) s(x, z) m(dz)  = ϕ0 (x) (u(x + z) − u(x) − z · ∇u(x)) s(x, z) m(dz) $

RN \{0}

+ RN \{0}



%

(u(x + z) − u(x))z s(x, z) m(dz)

· ∇ϕ0 (x)

(ϕ0 (x + z) − ϕ0 (x) − z · ∇ϕ0 (x)) u(x + z) s(x, z) m(dz)

+ RN \{0}

$

= ϕ0 (x)Su(x) + 

RN \{0}

% (u(x + z) − u(x))z s(x, z) m(dz)

· ∇ϕ0 (x)

(ϕ0 (x + z) − ϕ0 (x) − z · ∇ϕ0 (x)) u(x + z) s(x, z) m(dz).

+ RN \{0}

Hence we can write the commutator [S, ϕ0 ]u in the following form: [S, ϕ0 ] u(x) (1)

(2)

:= S0 u(x) + S0 u(x) $ = RN \{0}

%

(u(x + z) − u(x))z s(x, z) m(dz)

· ∇ϕ0 (x)

 +

RN \{0}

(ϕ0 (x + z) − ϕ0 (x) − z · ∇ϕ0 (x)) u(x + z) s(x, z) m(dz). (1)

First, just as in Lemma 14.4 we can estimate the term S0 u as follows:      (1)   (1)  = S0 u p  S0 u p L (D) L (G )   ≤ 2 σ(η) u C 1 (D) + δ(η) u C(D) ∇ϕ0 Lp (G )     C1 + C2 u C(D) ∇ϕ0 Lp (G ) . ≤ 2 σ(η) u C 1 (D) + η However, it follows from an application of Claim 12.4 that • ∇ϕ0 Lp (G ) ≤ Cη N/p ∇ϕ0 C(G ) ≤ C  η −1+N/p ,     • ∇2 ϕ0 Lp (G ) ≤ Cη N/p ∇2 ϕ0 C(G ) ≤ C  η −2+N/p ,

14.4 Proof of Theorem 1.10

425

since we have, as η ↓ 0,   |∇ϕ0 | = O η −1 ,

 2    ∇ ϕ0  = O η −2 .

Hence we obtain that      (1)  ≤ C η −1+N/p u C 1 (D) + η −2+N/p u C(D) . S0 u p L (D)

(14.24)

Similarly, by arguing as in the proof of Lemma 14.15 we can estimate the (2) term S0 u as follows:      (2)  ≤ C u C(D) ∇2 ϕ0 Lp (G ) (14.25) S0 u p L (D)   ≤ C u C(D) η N/p ∇2 ϕ0 C(G ) ≤ Cη −2+N/p u C(D) . Since we have the formula (1)

(2)

[S, ϕ0 ] u = S0 u + S0 u, the desired estimate (14.23b) follows by combining estimates (14.24) and (14.25). The proof of Claim 14.14 is complete.   Therefore, by combining four estimates (14.21), (14.22), (14.23a) and (14.23b) we obtain that |λ|

1/2

u C 1 (G ) + |λ| u C(G )

≤ Cp (ε) |λ|

N/2p

= Cp (ε) |λ|

N/2p

(14.26)

(W − λ)(ϕ0 u) Lp (D)

ϕ0 ((W − λ)u) + [A, ϕ0 ] u + [S, ϕ0 ] u Lp (D)  N/2p ≤ C |λ| η N/p (W − λ)u C(G ) + η −1+N/p u C 1 (G )  −2+N/p +η

u C(G )  N/2p η N/p (W − λ)u C(D) + η −1+N/p u C 1 (D) ≤ C |λ|  −2+N/p +η

u C(D) for all u ∈ D (W). Here and in the following the letter C denotes a generic positive constant depending on p and ε, but independent of u and λ. (3-b) We recall (see Figure 12.3) that the closure D = D ∪ ∂D can be covered by a finite number of sets of the forms

426

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11



for x0 ∈ D, for x0 ∈ ∂D.

B(x0 , η/2) B(x0 , η/2) ∩ D

Therefore, by taking the supremum of inequality (14.26) over x ∈ D we find that |λ|1/2 u C 1 (D) + |λ| u C(D) (14.27)   N/2p N/p

(W − λ)u C(D) + η −1 u C 1 (D) + η −2 u C(D) ≤ C |λ| η for all u ∈ D (W). (4) We now choose the localization parameter η. We let η0 K, |λ|1/2

η=

where K is a positive constant (to be chosen later on) satisfying the condition 0 0, there exists a point x ∈ D \ M such that  u(x) = maxD u, Au(x) ≤ 0.   (c) For all α > 0, the range R (αI − A) is dense in C0 D \ M .   Then the operator A is closable in the space C0 D \ M , and its minimal closed extension A generates a Feller semigroup {Tt }t≥0 on the state space D \ M. Proof. We apply part (ii) of Theorem 3.36 (the Hille–Yosida–Ray theorem) to the linear operator A. To do this, we have only to prove the following five assertions (1) through (5):   (1) The operator A is closable in the space C0 D \ M .

14.5 Proof of Theorem 1.11

429

  (2) R(αI − A) = C0 D \ M for α > 0.   (3) If u ∈ D A takes a positive maximum at a point x of K, then we have the inequality Au(x ) ≤ 0. 0α := (αI − A)−1 for α > 0. (4) G   0 α f ≤ 1 max f on D. (5) f ∈ C0 D \ M , f ≥ 0 on D =⇒ 0 ≤ G D α The proof is divided into six steps. Step 1: First, we prove that, for all α > 0 u ∈ D (A) , (α − A) u = f ≥ 0 1 =⇒ 0 ≤ u ≤ max f on D. α D

on D

(14.31)

Our proof is based on a reduction to absurdity. If we assume, to the contrary, that min u < 0, D

then it follows from condition (b) with u := −u that there exists a point x0 ∈ D \ M such that  −u(x0 ) = maxD (−u) = − minD u > 0, −Au(x0 ) ≤ 0. Hence we have the assertion 0 ≤ f (x0 ) = αu(x0 ) − Au(x0 ) < 0. This contradiction proves that u ≥ 0 on D. Similarly, if we assume, to the contrary, that max u > D

1 max f, α D

then we can find a point x1 ∈ D \ M such that ⎧ ⎨u(x ) = max u > 1 max f, 1 D D α ⎩Au(x ) ≤ 0. 1

Hence it follows that f (x1 ) = α u(x1 ) − Au(x1 ) ≥ αu(x1 ) > max f. D

430

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

This contradiction proves that max u ≤ D

1 max f. α D

Step 2: Secondly, by virtue of assertion (14.31) we can define an inverse −1

Gα := (αI − A)

: R (αI − A) −→ D (A)

which is non-negative and bounded on the range R (αI − A): 1 max f on D. α D   Furthermore, since the range R(αI − A) is dense in C0 D \ M , we find that the operator Gα can be uniquely extended  to a non-negative, bounded linear 0 α (see Figure 14.1 operator on the whole space C0 D \ M , denoted by G below). f ∈ R (αI − A) , f ≥ 0

on D =⇒ 0 ≤ Gα f ≤

G

−−−α−→

C0 (D \ M ) ⏐ ⏐

C0 (D \ M ) ⏐ ⏐

R (αI − A) −−−−−−→ (αI−A)−1

D(A)

α Fig. 14.1. The operators (αI − A)−1 and G

More precisely, it should be noticed that   f ∈ C0 D \ M , f (x) ≥ 0 on D 0 α f (x) ≤ 1 max f on D, =⇒ 0 ≤ G α D

(14.32)

and that

1 for all α > 0. α Step 3: Thirdly, we prove that       0 lim αG u − u  = 0 for each u ∈ C0 D \ M . α 0α ≤

G

α→∞

∞

Here

u ∞ = sup |u(x)| = max |u(x)| . x∈D

x∈D

(14.33)

(14.34)

14.5 Proof of Theorem 1.11

431

For any given ε > 0, we can find a function v ∈ D(A) such that

u − v < ε. Then we have, by assertion (14.33),        0  0   0  + αG + v − u ∞ αGα u − u ≤ α G α (u − v) α v − v ∞ ∞ ∞    0  ≤ 2 u − v ∞ + αG α v − v ∞    0  ≤ 2ε + αGα v − v  .

(14.35)

∞

However, it should be noticed that 0α (αI − A) v = αG 0α v − G 0 α (Av) . v=G Hence we have, by assertion (14.33),    0  αGα v − v 

∞

  0  = G α (Av)

∞

≤

1

Av ∞ . α

(14.36)

Therefore, by combining inequalities (14.35) and (14.36) we obtain that     0 αGα u − u

∞

so that

≤ 2ε +

1

Av ∞ , α

    0 u − u lim sup αG  α α→∞

∞

≤ 2ε.

This proves the desired assertion (14.34), since ε is arbitrary. Step 4: Now we can prove that the operator A is closable in the space C0 D \ M . To do this, we assume that  {un } ⊂ D(A), un −→ 0,   Aun −→ v in C0 D \ M . Then it follows that, for all α > 0, (αI − A) un = αun − Aun −→ −v

  in C0 D \ M .

0α , Moreover, we have, by the boundedness of G 0α (αI − A) un −→ −Gα v un = Gα (αI − A) un = G Hence we have the assertion Gα v = 0

for all α > 0.

  in C0 D \ M .

432

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

Therefore, we obtain from assertion (14.34) that 0α v = 0. v = lim αG α→∞

  This proves that A is closable in C0 D \ M . Step 5:  Let A be the minimal closed extension of A in the space C0 D \ M . Finally, it remains to prove the formula   0α = αI − A −1 G (a) First, we prove the formula   0 α αI − A = I G

for all α > 0.

(14.37)

on D(A).

(14.38)

Let u be an arbitrary element of D(A). Then we can find a sequence {un } in D(A) such that    in C0 D \ M , un −→ u   Aun −→ Au in C0 D \ M . 0α is bounded, it follows that Since G 0 α (αI − A) un = αG 0 α un − G 0α (Aun ) un = G     0 α Au 0α u − G in C0 D \ M . −→ αG Hence we have the formula   0 α αI − A u, u=G so that

  0 α αI − A u = u. G

This proves formula (14.38). (b) Next, we prove the formula   0α = I αI − A G

  on C0 D \ M . (14.39)   Let f be an arbitrary element of the space C0 D \ M . Since the range R (αI − A) is dense, we can find a sequence {un } in D (A) such that   (α − A) un −→ f in C0 D \ M . Then we have the assertions 0 α (α − A) un −→ G 0α f un = G and

  in C0 D \ M

14.5 Proof of Theorem 1.11

0α f Aun = (A − α) un + α un −→ −f + αG

433

  in C0 D \ M .

Since A is closed, it follows that  0 α f ∈ D(A), G   0α f = −f + αG 0 α f, A G or equivalently,

  0α f = f. αI − A G

This proves formula (14.39). Step 6: By virtue of assertions (14.37), (14.32) and (14.33), we obtain from part (ii) of Theorem 4.13 that the minimal closed extension A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on the state space D \ M. The proof of Theorem 14.16 is complete.   End of Proof of Theorem 1.11: We have only to verify conditions (a), (b) and (c) in Theorem 14.16 for the operator W defined by formula (1.17). Condition (b): We show that, for each α > 0 the equation (αI − W) u = f   has a unique solution u ∈ D (W) for any f ∈ C θ0 (D) ∩ C0 D \ M . Remark that the space   C θ0 (D) ∩ C0 D \ M is dense in C0 (D \ M ). Since we have the inequality c(x) − α ≤ −α for all x ∈ D, by applying Theorem 1.8 to the operator W − α we obtain that the boundary value problem  (α − W ) u = f in D, Lu = 0 on ∂D has a unique solution   u ∈ C 2+θ0 (D) ∩ C0 D \ M for any f ∈ C θ0 (D). However, we have the assertion ∂u  (x ) + γ(x )u(x ) = 0 on ∂D ∂n =⇒ u = 0 on M = {x ∈ ∂D : μ(x ) = 0}   =⇒ u ∈ C0 D \ M . Lu = μ(x )

434

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

  Therefore, if f ∈ C θ0 (D) ∩ C0 D \ M then it follows that   W u = αu − f ∈ C θ0 (D) ∩ C0 D \ M . This proves that



u ∈ D (W) , (αI − W) u = f.

Condition (c): We show that, for each α > 0 the Green operator     Gα := (αI − W)−1 : C0 D \ M ∩ C θ0 (D) −→ C0 D \ M ∩ C 2+θ0 (D) is non-negative: f ∈ C θ0 (D) ∩ C0 (D \ M ), f (x) ≥ 0 in D =⇒ u(x) = Gα f (x) ≥ 0 in D. Indeed, if we let v(x) = −u(x) = −Gα f (x), then it follows that



(W − α) v = f ≥ 0 Lv = 0

in D, on ∂D.

Therefore, by applying Proposition 14.7 to the operator W − α we obtain that v(x) ≤ 0 in D. This proves that u(x) = −v(x) ≥ 0 in D. Condition (a): In order to verify condition (a), we show that, for each α > 0 the Green operator Gα = (αI − W)−1   is bounded on the space C θ0 (D) ∩ C0 D \ M with norm 1/α:

Gα ≤

1 α

for all α > 0.

(14.40)

  Let f (x) be an arbitrary function in C θ0 (D) ∩ C0 D \ M . If we let u± (x) := ±αGα f (x) − f ∞ ∈ C 2+θ0 (D), then we have only to prove that u± (x) ≤ 0 in D.

(14.41)

14.5 Proof of Theorem 1.11

435

Indeed, it follows that (W − α) u± (x) = ±αf (x) + (α − c(x)) f ∞ = α ( f ∞ ± f (x)) + (α − c(x)) f ∞ ≥ 0 in D, and further that L (u± ) = −L ( f ∞ ) = −γ(x ) f ∞ ≥ 0

on ∂D.

Therefore, the desired assertion (14.41) follows by applying Proposition 14.7 to the operator W − α.   Now we show that the domain D (W) is dense in C0 D \ M . More pre  cisely, we prove that, for each u ∈ C 2+θ0 (D) ∩ C0 D \ M we have the assertion lim αGα u − u ∞ = 0. (14.42) α→+∞

Remark that the space   C 2+θ0 (D) ∩ C0 D \ M is dense in C0 (D \ M ). First, by Lemma 14.5 it follows that Su ∈ C θ0 (D) for u ∈ C 2+θ0 (D). Hence we have the assertions W u = Au + Su ∈ C θ0 (D) and so

for u ∈ C 2+θ0 (D),

  Gα (W u) ∈ C 2+θ0 (D) ∩ C0 D \ M .

If we let   w := α Gα u − Gα (W u) ∈ C 2+θ0 (D) ∩ C0 D \ M , then it follows that (W − α) w = −αu + W u = (W − α) u in D. Hence we have the assertions    w − u ∈ C 2+θ0 (D) ∩ C0 D \ M , (W − α) (w − u) = 0 in D. By applying Theorem 1.8 to the operator W − α, we obtain that w − u = 0 in D,

436

14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11

that is, u = w = αGα u − Gα (W u). Therefore, the desired assertion (14.42) follows from an application of inequality (14.40): lim sup u − αGα u ∞ = lim sup Gα (W u) ∞ ≤ lim α→+∞

α→+∞

α→+∞

1 · W u ∞ α

= 0. Now the proof of Theorem 1.11 is complete.  

14.6 Minimal Closed Extension W By combining Theorems 1.11 and 1.10, we can prove that the operator W coincides with the minimal closed extension W: W = W.

(14.43)

Indeed, we recall that • D (W)       = u ∈ C 2 (D) ∩ C0 D \ M : W u ∈ C0 D \ M , Lu = 0 on ∂D , • D (W)       = u ∈ H 2,p (D) ∩ C0 D \ M : W u ∈ C0 D \ M , Lu = 0 on ∂D . Then we have, by Theorems 1.11 and 1.10, W ⊂ W. Therefore, we have only to show that   D W = D (W) .

(14.44)

(14.45)

Let u be an arbitrary element of D (W). Since the operators W and W  generate semigroups of class (C0 ) on the space C0 D \ M , it follows from an application of part (i) of Theorem 3.34 (the Hille–Yosida theorem) with α := 1 that the operators     I − W : D W −→ C0 D \ M and

  I − W : D (W) −→ C0 D \ M

  are both bijective. Hence we can find an (unique) element v ∈ D W such that

14.7 Notes and Comments

437

  I − W v = (I − W) u. However, by assertion (14.44) it follows that  v − u ∈ D (W) , (I − W) (v − u) = 0, so that

  u=v∈D W .

This proves assertion (14.45) and hence the desired assertion (14.43). Summing up, we have proved the following theorem: Theorem 14.17. If conditions (A), (B) and (H) are satisfied, then the closed operator W is the infinitesimal generator of some Feller semigroup {et W }t≥0 on the state space D \ M . Moreover, it generates a semigroup ez W on the Banach space C0 D \ M that is analytic in the sector Δε = {z = t + is : z = 0, |arg z| < π/2 − ε} for any 0 < ε < π/2 (see Figure 1.3).

14.7 Notes and Comments The results discussed in this chapter are adapted from Taira [117], [121], [123]. [127] and [128]. Gohberg–Kre˘ın [51] is the classic for index theory of closed linear operators in Banach spaces.

15 Path Functions of Markov Processes via Semigroup Theory

In this book we have studied mainly Markov transition functions with only informal references to the random variables which actually form the Markov processes themselves (see Section 3.3). In this chapter we study this neglected side of our subject. The discussion will have a more measure-theoretical flavor than hitherto. Section 15.1 is devoted to a review of the basic definitions and properties of Markov processes. In Section 15.2 we consider when the paths of a Markov process are actually continuous, and prove Theorem 3.19 (Corollary 15.7). In Section 15.3 we give a useful criterion for path-continuity of a Markov process {xt } in terms of the infinitesimal generator A of the associated Feller semigroup {Tt } (Theorem 15.9). Section 15.4 is devoted to the study of three typical examples of multi-dimensional diffusion processes. More precisely, we prove that (1) the reflecting barrier Brownian motion (Theorem 15.11), (2) the reflecting and absorbing barrier Brownian motion (Theorem 15.14) and (3) the reflecting, absorbing and drifting barrier Brownian motion (Theorem 15.15) are multi-dimensional diffusion processes, namely, they are continuous strong Markov processes. This chapter is taken from [122, Chapter 12].

15.1 Basic Definitions and Properties of Markov Processes First, we recall the basic definitions of stochastic processes (see Subsection 3.2.1). Let K be a locally compact, separable metric space and B the σalgebra of all Borel sets in K. Let (Ω, F , P ) be a probability space. A function X defined on Ω taking values in K is called a random variable if it satisfies the condition X −1 (E) = {X ∈ E} ∈ F for all E ∈ B, that is, X is F /B-measurable. © Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 15

440

15 Path Functions of Markov Processes via Semigroup Theory

A family X = {x(t, ω)}, t ∈ [0, ∞), ω ∈ Ω, of random variables is called a stochastic process. We regard the process X primarily as a function of t whose values x(t, ·) for each t are random variables defined on Ω taking values in K. More precisely, we are dealing with one function of two variables, that is, for each fixed t the function xt (·) is F /B-measurable. If, instead of t, we fix an ω ∈ Ω, then we obtain a function x(·, ω) : [0, ∞) → K which may be thought of as the motion in time of a physical particle. In this context, the space K is called the state space and Ω the sample space. The function xt (ω) = x(t, ω), t ∈ [0, ∞), defines in the state space K a trajectory or a path of the process corresponding to the sample point ω. Sometimes it is useful to think of a stochastic process X specifically as a function of two variables x(t, ω) where t ∈ [0, ∞) and ω ∈ Ω. One powerful tool in this connection is Fubini’s theorem. To do this, we introduce a class of stochastic processes which we will deal with in this appendix. Definition 15.1. A stochastic process X = {xt }t≥0 is said to be measurable provided that the function x(·, ·) : [0, ∞) × Ω → K is measurable with respect to the product σ-algebra A × F , where A is the σ-algebra of all Borel sets in the interval [0, ∞). We remark that the condition that the function xt (·) = x(t, ·) is F /B-measurable for each t does not guarantee the measurability of the process x(·, ·). Now let pt be a Markov transition function on the metric space K (see Definition 3.17). The idea behind Definition 3.17 of a Markov transition function suggests the following definition (cf. formula (3.17)): Definition 15.2. A stochastic process X = {xt }t≥0 is said to be governed by the transition function pt provided that we have, for all 0 ≤ t1 < t2 < . . . < tn < ∞ and all Borel sets B1 , B2 , . . ., Bn ∈ B, P (ω ∈ Ω : xt1 (ω) ∈ B1 , xt2 (ω) ∈ B2 , . . . , xtn (ω) ∈ Bn )     = ... μ(dx) pt1 (x, dy1 ) yn ∈Bn

y1 ∈B1

y1 ∈B1

(15.1)

x∈K

pt2 −t1 (y1 , dy2 ) · · · ptn −tn−1 (yn−1 , dyn ), where μ is some probability measure on the measurable space (K, B), and is called the initial distribution of the process {xt }. The notion of the Markov property is introduced and discussed in Chapter 3, Section 3.2: If X = {xt }t≥0 is a stochastic process, we introduce three subσ-algebras of F as follows:

15.1 Basic Definitions and Properties of Markov Processes

⎧ F≤t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨F=t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪F≥t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

441

:= σ(xs : 0 ≤ s ≤ t) = the smallest σ-algebra, contained in F with respect to which all xs , 0 ≤ s ≤ t, are measurable, := σ(xt ) = the smallest σ-algebra, contained in F with respect to which xt is measurable, := σ(xs : t ≤ s < ∞) = the smallest σ-algebra, contained in F with respect to which all xs , t ≤ s < ∞, are measurable.

We recall that an event in F≤t is determined by the behavior of the process {xs } up to time t and an event in F≥t by its behavior after time t. Thus they represent respectively the “past” and “future” relative to the “present” moment. A stochastic process X = {xt } is called a Markov process if it satisfies the following condition: P (B | F≤t ) = P (B | F=t )

for any “future” set B ∈ F≥t .

More precisely, we have, for any “future” set B ∈ F≥t ,  P (B | F=t )(ω) dP (ω) for every “past” set A ∈ F≤t . P (A ∩ B) = A

Intuitively, this means that the conditional probability of a “future” event B given the “present” is the same as the conditional probability of B given the “present” and “past”. The next theorem justifies Definition 15.2, and hence it is fundamental for our further study of Markov processes: Theorem 15.3. Let X = {xt }t≥0 be any stochastic process with values in the metric space K which is governed by a Markov transition function pt . Then it follows that {xt } is a Markov process. Our study of Markov processes is based on formula (15.1) that shows how the finite-dimensional distributions of the process X = {xt } are calculated from the Markov transition function pt . However, knowledge of all the finitedimensional distributions may not be sufficient to determine precisely the path functions of a Markov process. Therefore, it is important to ask the following question: Question 15.4. Given a Markov transition function pt and an initial distribution μ, does there exist a Markov process X = {xt } having the corresponding finite-dimensional distributions whose paths are almost surely “nice” in some sense ?

442

15 Path Functions of Markov Processes via Semigroup Theory

We say that two Markov processes X = {xt }t≥0 and Y = {yt }t≥0 defined on the same probability space (Ω, F , P ) is equivalent provided that we have, for all t ∈ [0, ∞), P ({ω ∈ Ω : xt (ω) = yt (ω)}) = 1. The next theorem asserts that, under quite general conditions there does exist a Markov process with “nice” paths equivalent to any given process: Theorem 15.5. Let (K, ρ) be a compact metric space and let X = {xt }t≥0 be a stochastic process with values in K which is governed by a normal Markov transition function pt . Then there exists a Markov process Y = {yt }, equivalent to the process X = {xt }, such that P ({ω ∈ Ω : the function yt (ω) are right-continuous and have left-hand limits for all t ≥ 0}) = 1.

15.2 Path-Continuity of Markov Processes It is naturally interesting and important to consider when the paths of a Markov process {xt } are actually continuous for all t ≥ 0. The purpose of this section is to establish some useful sufficient conditions for path-continuity of the Markov process {xt }. First, we have the following theorem: Theorem 15.6. Let (K, ρ) be a locally compact metric space and let X = {xt }t≥0 be a measurable stochastic process with values in K. Assume that, for each ε > 0 and each M > 0 the condition P ({ω ∈ Ω : ρ(xt (ω), ρt+h (ω)) ≥ ε}) = o(h)

(15.2)

holds true uniformly for all t ∈ [0, M ], as h ↓ 0. Namely, we have, for all t ∈ [0, M ], P ({ω ∈ Ω : ρ(xt (ω), ρt+h (ω)) ≥ ε}) lim = 0. h↓0 h Then it follows that P ({ω ∈ Ω : xt (ω) has a jump discontinuity somewhere}) = 0.

(15.3)

Proof. The proof is divided into two steps. Step 1: First, we show that if we let Jε,h (ω) = {t ∈ [0, M ] : ρ(xt (ω), xt+h (ω)) ≥ ε},

ω ∈ Ω,

(15.4)

15.2 Path-Continuity of Markov Processes

443

then it follows from condition (15.2) that E [m(Jε,h )] = o(h)

as h ↓ 0,

(15.5)

where m = dt is the Lebesgue measure on R. To do this, we define the set (see Figure 15.1 below) Aε,h = {(t, ω) ∈ [0, M ] × Ω : ρ(xt (ω), xt+h (ω)) ≥ ε} . We remark that the set Aε,h is measurable with respect to the product σΩ

Aε,h

ω

t

0

M

Fig. 15.1. The set Aε,h

algebra A × F. Indeed, it suffices to note the following two facts (a) and (b): (a) The mapping (t, ω) → (xt (ω), xt+h (ω)) is measurable from the product space [0, M ] × Ω into the product space K × K, for each h ≥ 0. (b) The metric ρ is a continuous function on the product space K × K. By virtue of Fubini’s theorem, we can compute the product measure m×P of Aε,h by integrating the measure of a cross section as follows:  M (m × P )(Aε,h ) = P ({ω ∈ Ω : (t, ω) ∈ Aε,h }) dt. 0

By condition (15.2), it follows that (m × P )(Aε,h ) = o(h) as h ↓ 0.

(15.6)

Moreover, by integrating in the other order we obtain from definition (15.4) that  (m × P )(Aε,h ) = m({t ∈ R : (t, ω) ∈ Aε,h }) dP (ω) (15.7) Ω

444

15 Path Functions of Markov Processes via Semigroup Theory

= E [m({t ∈ [0, M ] : ρ(xt (ω), xt+h (ω)) ≥ ε})] = E [m(Jε,h )] . Therefore, the desired assertion (15.5) follows by combining assertions 15.6 and (15.7). Step 2: Now we show that the existence of jumps in the trajectories of {xt } contradicts assertion (15.5). Step 2-1: We assume that, for some ε0 > 0, P ({ω ∈ Ω : xt (ω) has a jump with gap greater than 2ε0 }) > 0. If xt (ω) is a trajectory having such a jump at t = t0 (see Figure 15.2 below), then we obtain that the two limits xt+ (ω) = lim xt0 +h (ω), 0

xt− (ω) = lim xt0 −h (ω) 0

h↓0

h↓0

exist and satisfy the condition ρ(xt− (ω), xt+ (ω)) ≥ 2ε0 . 0

0

Then we have, for sufficiently small h > 0,

xs (ω)

ε0 ε0

t0 − h

s t0

t0 + h

Fig. 15.2. The condition ρ(xt− (ω), xt+ (ω)) ≥ 2ε0 0



ρ(xt− (ω), xs (ω)) ≤ 0

ρ(xt+ (ω), xs (ω)) ≤ 0

ε0 2 ε0 2

0

for all s ∈ (t0 − h, t0 ), for all s ∈ (t0 , t0 + h).

Moreover, by the triangle inequality it follows that 2ε0 ≤ ρ(xt− (ω), xt+ (ω)) 0

0

≤ ρ(xt− (ω), xt (ω)) + ρ(xt (ω), xt+h (ω)) + ρ(xt+h (ω), xt+ (ω)) 0

0

15.2 Path-Continuity of Markov Processes

≤

ε0 ε0 + ρ(xt (ω), xt+h (ω)) + 2 2

445

for all t ∈ (t0 − h, t0 ),

so that ρ(xt (ω), xt+h (ω)) ≥ ε0

for all t ∈ (t0 − h, t0 ).

This implies that, for all sufficiently small h > 0, m ({t ∈ [0, M ] : ρ(xt (ω), xt+h (ω)) ≥ ε0 }) ≥ m ((t0 − h, t0 )) = h, or equivalently, m(Jε0 ,h )(ω) ≥1 h

for all sufficiently small h > 0.

(15.8)

Step 2-2: The proof is based on a reduction to absurdity. We assume, to the contrary, that P ({ω ∈ Ω : xt (ω) has a jump discontinuity somewhere}) > 0. Then it follows from Step 2-1 that there exists a positive number ε0 such that assertion (15.8) holds true. Therefore, we can find a positive constant δ = δ(ε0 ) such that  E[m(Jε0 ,h )] = m(Jε0 ,h )(ω) dP (ω) (15.9) Ω  ≥ m(Jε0 ,h )(ω) dP (ω) L ε0

≥ hδ

for all sufficiently small h > 0,

where Lε0 = {ω ∈ Ω : xt (ω) has a jump with gap greater than 2ε0 } . Assertion (15.9) contradicts condition (15.5). The proof of Theorem 15.6 is complete.   The next corollary proves part (ii) of Theorem 3.19 under condition (N) with E = K: Corollary 15.7. Let (K, ρ) be a locally compact metric space and let X = {xt }t≥0 be a right-continuous Markov process governed by a Markov transition function pt . Assume that, for each ε > 0, the condition ph (x, K \ Uε (x)) = o(h)

(15.10)

holds true uniformly in x ∈ K as h ↓ 0. In other words, for each ε > 0 we have the condition

446

15 Path Functions of Markov Processes via Semigroup Theory

lim h↓0

1 sup pt (x, K \ Uε (x)) = 0. h x∈K

Here Uε (x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x. Then it follows that P ({ω ∈ Ω : xt (ω) is continuous for all t ≥ 0}) = 1.

Proof. We shall apply Theorem 15.6 to our situation. It should be emphasized that the stochastic process {xt } is right-continuous and has limits from the left as well. The proof of Corollary 15.7 is divided into two steps. Step 1: First, we prove the following lemma (see [33, Lemma 5.9]): Lemma 15.8. Every right-continuous stochastic process {xt } is measurable. Proof. If {tn } is an increasing sequence such that 0 = t0 < t1 < t2 < . . . → ∞, then we define a stochastic process {yt } by the formula (see Figure 15.3 below) ⎧ ⎪ ⎪xt0 (ω) = x(t0 , ω), ω ∈ Ω ⎪ ⎪ ⎪ xt1 (ω) = x(t1 , ω), ω ∈ Ω ⎪ ⎪ ⎪ ⎪ ⎪ · ⎨ yt (ω) = · ⎪ ⎪ ⎪ xtk (ω) = x(tk , ω), ω ∈ Ω ⎪ ⎪ ⎪ ⎪ ⎪ · ⎪ ⎪ ⎩ ·

for t0 ≤ t < t1 , for t1 ≤ t < t2 ,

for tk ≤ t < tk+1 ,

t t0 = 0

t1

t2

tk

tk+1

Fig. 15.3. The stochastic process {yt }

Since we have, for any a ∈ R,

15.2 Path-Continuity of Markov Processes

{(t, ω) : yt (ω) < a} =

∞ "

447

[ti , ti+1 ) × {ω : xti (ω) < a},

i=0

it follows that the process {yt } is measurable. For each integer n ∈ N, we choose a non-negative integer k such that k+1 k ≤t< , n 2 2n and let

k+1 . 2n It is clear that φn (t) ↓ t as n → ∞. Now, if we define a stochastic process {xn } by the formula φn (t) =

xn (t, ω) = x(φn (t), ω) for every ω ∈ Ω, then we have the following two assertions (a) and (b): (a) The process xn (t) is measurable. (b) xn (t, ω) → x(t, ω) as n → ∞. Indeed, assertion (a) is proved just as in the case of the process {yt }, while assertion (b) follows from the right-continuity of the process {xt }. Therefore, we obtain from assertions (a) and (b) that the process {xt } is measurable. The proof of Lemma 15.8 is complete.   Step 2: Let μ be the initial distribution of the process {xt } in Definition 15.2. Namely, we have, for all 0 ≤ t1 < t2 < . . . < tn < ∞ and all Borel sets B1 , B2 , . . ., Bn ∈ B, P (ω ∈ Ω : xt1 (ω) ∈ B1 , xt2 (ω) ∈ B2 , . . . , xtn (ω) ∈ Bn )     = ... μ(dx)pt1 (x, dy1 ) yn ∈Bn

y1 ∈B1

y1 ∈B1

x∈K

pt2 −t1 (y1 , dy2 ) · · · ptn −tn−1 (yn−1 , dyn ). Then we have, by Fubini’s theorem (see Figure 15.4 below), P ({ω ∈ Ω : ρ(xt (ω), xt+h (ω)) ≥ ε}) = P ({ω ∈ Ω : xt (ω) ∈ K, xt+h (ω) ∈ K \ Uε (xt (ω))})     pt (x, dy) ph (y, dz) μ(dx) = K

K\Uε(y)

  $

K

%

ph (y, dz) pt (x, dy) μ(dx)

= K

K\Uε(y)

(15.11)

448

15 Path Functions of Markov Processes via Semigroup Theory

xt+h (ω) = z

Uε (xt (ω)) xt (ω) = y

x Fig. 15.4. The proof of formula (15.11)

 pt (x, dy)ph (y, K \ Uε (y)) μ(dx).

= K

In view of condition 15.10, we obtain from formula (15.11) that P ({ω ∈ Ω : ρ(xt (ω), xt+h (ω)) ≥ ε}) h ++ pt (x, dy)ph (y, K \ Uε (y)) μ(dx) = lim K h↓0 h    ph (y, K \ Uε (y)) = pt (x, dy) lim μ(dx) h↓0 h K lim h↓0

=0 holds true uniformly for all t ≥ 0. This assertion implies that condition (15.2) holds true uniformly for all t ≥ 0, as h ↓ 0. Hence it follows from an application of Theorem 15.6 that P ({ω : xt (ω) has a jump discontinuity somewhere}) = 0.

(15.12)

However, we recall that the stochastic process {xt } is right-continuous and has limits from the left as well. Therefore, we obtain from assertion (15.12) that P ({ω ∈ Ω : xt (ω) is continuous for all t ≥ 0}) = 1. The proof of Corollary 15.7 is complete.  

15.3 Path-Continuity of Feller Semigroups

449

15.3 Path-Continuity of Feller Semigroups It is usually difficult to verify condition 15.10 directly, since it is rather exceptional when any simple formula for the transition probability function pt is available. The purpose of this section is to give a useful criterion for pathcontinuity of the Markov process {xt } in terms of the infinitesimal generator A of the associated Feller semigroup {Tt }. Let (K, ρ) be a compact metric space and let C(K) be the space of realvalued, bounded continuous functions on K; C(K) is a Banach space with the maximum norm

f ∞ = max |f (x)|. x∈K

A strongly continuous semigroup {Tt }t≥0 on the compact metric space C(K) is called a Feller semigroup if it is non-negative and contractive on C(K), that is, f ∈ C(K), 0 ≤ f (x) ≤ 1 on K =⇒ 0 ≤ Tt f (x) ≤ 1 on K. We recall (see Theorem 3.31) that if pt is a uniformly stochastically continuous, Feller function on K, then its associated operators {Tt }t≥0 , defined by formula  pt (x, dy)f (y) for every f ∈ C(K), (15.13) Tt f (x) = K

form a Feller semigroup on K. Conversely, if {Tt }t≥0 is a Feller semigroup on K, then there exists a uniformly stochastically continuous, Feller function pt on K such that formula (15.13) holds true. Indeed, it suffices to note that if K is compact, then condition (L) is trivially satisfied. Furthermore, we know that the function pt is the transition function of some strong Markov process X = {xt }t≥0 whose paths are right-continuous and have no discontinuities other than jumps. Our approach can be visualized as in Figure 15.5 below.

Feller semigroup on C(K)

uniform stochastic continuity

right-continuous Markov process

+

Feller property

strong Markov process

Fig. 15.5. A functional analytic approach to strong Markov processes

450

15 Path Functions of Markov Processes via Semigroup Theory

The next theorem gives some useful sufficient conditions for pathcontinuity of the Markov process {xt } in terms of the infinitesimal generator A of the associated Feller semigroup {Tt }: Theorem 15.9. Let (K, ρ) be a compact metric space and let X = {xt }t≥0 be a right-continuous Markov process governed by a uniformly stochastically continuous, Feller transition function pt . Assume that the infinitesimal generator A of the associated Feller semigroup {Tt }t≥0 , defined by formula (15.13), satisfies the following three conditions (15.14a), (15.14b) and (15.14c): For each ε > 0 and each point x ∈ K, there exists a function f ∈ D(A) such that f (x) ≥ 0 on K. f (y) > 0 for all y ∈ K \ Uε (x).

(15.14a) (15.14b)

f (y) = Af (y) = 0 in some neighborhood of x.

(15.14c)

Here Uε (x) = {z ∈ K : ρ(z, x) < ε} is an ε-neighborhood of x. Then it follows that P ({ω ∈ Ω : xt (ω) is continuous for all t ≥ 0}) = 1.

Proof. We shall apply Corollary 15.7 to our situation. The proof is based on a reduction to absurdity. We assume, to the contrary, that condition (15.10) does not hold true. Then we can find a positive number ε0 such that ph (x, K \ U2ε0 (x))

is not of order o(h) uniformly in x as h ↓ 0.

More precisely, for this ε0 there exist a positive constant δ, a decreasing sequence {hn } of positive numbers, hn ↓ 0, and a sequence {xn } of points in K such that phn (xn , K \ U2ε0 (xn )) ≥ δ hn

for all sufficiently large n.

(15.15)

However, since K is compact, we may assume that the sequence {xn } itself converges to some point x0 of K. If ρ(xn , x0 ) < ε0 , then it follows that (see Figure 15.6 below) Uε0 (xn ) ⊂ U2ε0 (x0 ). Hence we have, by assertion (15.15), phn (xn , K \ Uε0 (x0 )) ≥ phn (xn , K \ U2ε0 (xn )) ≥ δ hn for all sufficiently large n.

(15.16)

Now, for these ε0 and x0 we can construct a function f ∈ D(A) which satisfies

15.3 Path-Continuity of Feller Semigroups

451

xn x0

ε0

2ε0

Fig. 15.6. The condition Uε0 (xn ) ⊂ U2ε0 (x0 )

conditions (15.14). Then it follows from condition (15.14b) that c=

min

f (x) > 0.

x∈K\Uε0

Hence we have, by assertion (15.16),  Thn f (xn ) = phn (xn , dy) f (y) K ≥ phn (xn , dy) f (y) K\Uε0 (x0 )

 ≥

min

K\Uε0 (x0 )

≥ c δ hn

(15.17)

 f

· phn (xn , K \ Uε0 (x0 ))

for all sufficiently large n.

However, since f (xn ) = 0 for xn close to the limit point x0 , we obtain from assertion (15.16) that Th f (xn ) Thn f (xn ) − f (xn ) = n ≥ cδ hn hn

for all xn ∈ U (x0 ).

(15.18)

On the other hand, since we have, for f ∈ D(A), Thn f (y) − f (y) n→∞ hn

Af (y) = lim

uniformly in y ∈ K,

it follows from condition (15.14c) that Thn f (y) − f (y) −→ Af (y) = 0 uniformly in y ∈ U (x0 ). hn Hence we have, for all y ∈ U (x0 ),    Thn f (y) − f (y)  c δ <  for all sufficiently large n.   hn 2

(15.19)

452

15 Path Functions of Markov Processes via Semigroup Theory

However, since f (xn ) = 0 and Thn f ≥ 0 on K, by letting y := xn in inequality (15.19) we obtain that   Thn f (xn ) − f (xn )  Thn f (xn ) − f (xn )  cδ =  < 2 for all xn ∈ U (x0 ). hn hn This assertion contradicts assertion (15.18). Therefore, we have proved that condition (15.10) holds true. Theorem 15.9 follows from an application of Corollary 15.7. The proof of Theorem 15.9 is complete.  

15.4 Examples of Multi-dimensional Diffusion Processes In this section we prove that (1) the reflecting barrier Brownian motion, (2) the reflecting and absorbing barrier Brownian motion, (3) the reflecting, absorbing and drifting barrier Brownian motion are typical examples of multidimensional diffusion processes, that is, examples of continuous strong Markov processes. It should be emphasized that these three Brownian motions correspond to the Neumann boundary value problem, the Robin boundary value problem and the oblique derivative boundary value problem for the Laplacian Δ in terms of elliptic boundary value problems, respectively. Table 15.1 below gives a bird’s eye view of diffusion processes and elliptic boundary value problems and how these relate to each other: 15.4.1 The Neumann Case Let D be a bounded domain of Euclidean space RN with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary (see Figure 15.7 below).

∂D D

n

Fig. 15.7. The bounded domain D and the inward normal n to the boundary ∂D

First, we consider the Neumann boundary condition BN u =

∂u = 0 on ∂D, ∂n

15.4 Examples of Multi-dimensional Diffusion Processes

Diffusion Processes (Microscope approach)

Elliptic Boundary Value Problems (Mesoscopic approach)

Brownian motion

Laplacian Δ

Reflecting barrier Brownian motion

Neumann boundary condition BN

Reflecting and absorbing barrier Brownian motion

Robin boundary condition BR

Reflecting, absorbing and drift barrier Brownian motion

Oblique derivative boundary condition Bo

453

Table 15.1. A bird’s-eye view of diffusion processes and elliptic boundary value problems

where n = (n1 , n2 , . . . , nN ) is the unit inward normal to the boundary ∂D. We introduce a linear operator AN as follows: (a) The domain D(AN ) is the space   ∂u D(AN ) = u ∈ C(D) : Δu ∈ C(D), = 0 on ∂D . ∂n (b) AN u = Δu for every u ∈ D(AN ). ∂u Here Δu and ∂n are taken in the sense of distributions. Then it follows from an application of Theorem 1.6 with L := BN that the operator AN is the infinitesimal generator of a Feller semigroup {St }t≥0 . Let {xt (ω)} be the strong Markov process corresponding to the Feller semigroup {St }t≥0 with Neumann boundary condition. In this subsection we study the path-continuity of the Markov process {xt (ω)}. In order to make use of Theorem 15.9, we shall construct a function f ∈ D(AN ) which satisfies conditions (15.14) of the same theorem. Our construction of the function f (x) may be visualized as in Figure 15.8 below. (I) The case where x0 is an arbitrary (interior) point of D: By applying Theorem [121, Theorem 1.5] with μ(x ) := 1 and γ(x ) := 0, we can find a function φ ∈ C ∞ (D) such that

454

15 Path Functions of Markov Processes via Semigroup Theory f (y)

Uε (x)

∂D

∂D

U2ε (x)

Fig. 15.8. The function f ∈ D(AN )

 Δφ = −1 in D, ∂φ on ∂D. ∂n = 0

(15.20)

Then we have the following claim: Claim 15.10. The function satisfies the condition φ(x) > 0

on D.

Proof. The proof is based on a reduction to absurdity. We assume, to the contrary, that min φ ≤ 0. D

(i) First, we consider the case: There exists a point x0 ∈ D such that φ(x0 ) = min φ ≤ 0. D

Since we have, by condition (15.20), Δ(−φ) = 1 > 0 in D,

(15.21)

it follows from an application of the strong maximum principle (Theorem 10.7) with W := Δ and u := −φ that φ(x) ≡ φ(x0 ) in D. Hence we have the assertion Δ(−φ) = 0 in D. This contradicts condition (15.21). (ii) Next we consider the case: There exists a point x0 ∈ ∂D such that φ(x0 ) = min φ ≤ 0. D

Then, by applying the boundary point lemma (Lemma 10.11) with W := Δ to the function u := −φ we obtain from inequality (15.21) that

15.4 Examples of Multi-dimensional Diffusion Processes

455

∂φ  (x ) > 0. ∂n 0 This contradicts the boundary condition ∂φ = 0 on ∂D. ∂n The proof of Claim 15.10 is complete.   Now we choose a real-valued, smooth function θ ∈ C ∞ (R) such that (see Figure 15.9 below) ⎧ ⎪ on R, ⎨0 ≤ θ(t) ≤ 1 (15.22) supp θ ⊂ (−1, 1), ⎪  1 1 ⎩ θ(t) = 1 for all t ∈ − 2 , 2 , and define a function f ∈ C ∞ (D) by the formula    |x − x0 | f (x) = 1 − θ φ(x), ε where 0 < ε < dist (x, ∂D) = inf |x − z|. z∈∂D

1

−1

−1/2

0

θ(t)

1/2

1

t∈R

Fig. 15.9. The function θ ∈ C ∞ (R) that satisfies the conditions (15.22)

Then, in view of Claim 15.10 it is easy to verify that f ∈ D(AN ) and satisfies conditions (15.14) of Theorem 15.9. (II) The case where x0 is an arbitrary (boundary) point of ∂D: By change of coordinates, we may assume that  x0 = (0 , 0) = (0, . . . , 0, 0) , (15.23) n = (0 , 1) = (0, . . . , 0, 1) ∈ RN .

456

15 Path Functions of Markov Processes via Semigroup Theory xN

D = {xN > 0} 1

n = (0, . . . , 0, 1)

∂D = {xN = 0} x = (x1 , . . . , xN −1 ) x0 = 0

Fig. 15.10. The point x0 and the unit inward normal n in the formula (15.23)

The situation may be represented schematically as in Figure 15.10 above. If we define a function f ∈ C ∞ (D) by the formula   x   |x |  N θ f (x , xN ) = 1 − θ φ(x , xN ), ε ε x = (x1 , x2 , . . . , , xN −1 ) ∈ RN −1 , xN ∈ R, then it follows from conditions (15.22) and (15.20) that  ∂f  ∂f = ∂n ∂xN xN =0      ∂   xN  |x |  θ =− · φ(x , 0) θ ∂xN ε ε xN =0    x   |x |  ∂φ  N + 1−θ θ (x , xN ) · ε ε ∂xN xN =0        |x | |x ∂φ | 1  = − θ (0)θ · φ(x , 0) + 1 − φ ε ε ε ∂n =0

on ∂D.

This proves that f ∈ D(AN ). Moreover, we have the following three assertions (α), (β) and (γ) (see Figure 15.11 below): (α) Since f (x) = 0 in the ε/2 neighborhood Uε/2 (x0 ) of x0 , it follows that f (y) = AN f (y) = 0 for all y ∈ Uε/2 (x0 ). This proves that condition (15.14c) in Theorem 15.9 is satisfied.

15.4 Examples of Multi-dimensional Diffusion Processes

457

(β) It is clear that f (x) ≥ 0 on D, so√that condition (15.14a) is satisfied. (γ) Since f (y) = φ(y) outside the N ε neighborhood U√N ε (x0 ) of x0 , it follows from Claim 15.10 that f (y) > 0

for all y ∈ D \ U√N ε (x0 ).

This proves that condition (15.14b) is satisfied.

xN ε

ε/2

−ε

−ε/2

x

0

ε/2

ε

Fig. 15.11. The ε/2 neighborhood Uε/2 (x0 ) of x0 = 0

Therefore, by applying Theorem 15.9 to the operator AN we obtain the following theorem: Theorem 15.11 (the Neumann case). The strong Markov process {xt (ω)} associated with the Feller semigroup {St }t≥0 enjoys the property P ({ω : xt (ω) is continuous for all t ≥ 0}) = 1. Since {St }t≥0 is a Feller semigroup on the compact set D, it follows from an application of Theorem 3.31 that there exists a uniformly stochastically continuous, Feller transition function pt (x, ·) on D such that the formula  St f (x) = pt (x, dy)f (y) (15.24) D

holds true for all f ∈ C(D). Furthermore, we can prove the following important proposition: Proposition 15.12. The Feller transition function pt (x, ·) is conservative, that is, we have, for all t > 0, pt (x, D) = 1

for each x ∈ D.

(15.25)

458

15 Path Functions of Markov Processes via Semigroup Theory

Proof. First, we remark that 

1 ∈ D(AN ), AN 1 = 0 in D.

Hence, by applying the existence and uniqueness theorem for the initial-value problem with the semigroup St (cf. [115, Theorem 4.3]) we obtain that the function u(t) = St 1 is a unique solution of the initial-value problem  du dt = AN u for all t > 0, (∗) u(0) = 1 which satisfies the following three conditions (1), (2) and (3): (1) The function u(t) is continuously differentiable for all t > 0. (2) u(t) ≤ 1 for all t ≥ 0. (3) u(t) → 1 as t ↓ 0. However, it is easy to see that the function u(t) ≡ 1 is also a solution of problem (∗) which satisfies three conditions (1), (2) and (3). Therefore, it follows that St 1 = 1

for all t > 0.

In view of formula (15.24), we obtain the desired assertion (15.25) as follows:  1 = St 1(x) = pt (x, dy) = pt (x, D) for each x ∈ D. D

The proof of Proposition 15.12 is complete.   15.4.2 The Robin Case Secondly, we consider the Robin boundary condition BR u = μ(x ) where



∂u + γ(x )u = 0 ∂n

μ ∈ C ∞ (∂D), γ ∈ C ∞ (∂D),

μ(x ) > 0 γ(x ) ≤ 0

on ∂D,

(15.26)

on ∂D, on ∂D.

We introduce a linear operator AR as follows: (a) The domain D(AR ) is the space    ∂u  + γ(x )u = 0 on ∂D . D(AR ) = u ∈ C(D) : Δu ∈ C(D), μ(x ) ∂n

15.4 Examples of Multi-dimensional Diffusion Processes

459

(b) AR f = Δf for every f ∈ D(AR ). However, since μ(x ) > 0 on ∂D, by letting γ 0(x ) =

γ(x ) μ(x )

we find that the boundary condition 15.25 is equivalent to the following: ∂f +γ 0(x )f = 0 on ∂D. ∂n In other words, without loss of generality we may assume that μ(x ) = 1

on ∂D,



γ 0(x ) ≤ 0 on ∂D. Then it follows from an application of Theorem 1.6 with L := BR that the operator AR is the infinitesimal generator of a Feller semigroup {Tt }t≥0 with Robin boundary condition. Let {yt (ω)} be the strong Markov process corresponding to the Feller semigroup {Tt }t≥0 . In this subsection we study the path-continuity of the Markov process {yt (ω)}. To do this, we shall make use of Theorem 15.9. (I) The case where x0 is an arbitrary (interior) point of D: By applying Theorem [121, Theorem 1.5] with μ(x ) := 1 and γ(x ) := γ 0(x ), we can find ∞ a function φ ∈ C (D) such that  Δψ = −1 in D, (15.27) ∂ψ  γ (x )ψ = 0 on ∂D. ∂n + 0 Then we have the following claim: Claim 15.13. The function satisfies the condition ψ(x) > 0

on D.

Proof. The proof is based on a reduction to absurdity. We assume, to the contrary, that min ψ ≤ 0. D

(1) First, we consider the case: There exists a point x0 ∈ D such that ψ(x0 ) = min ψ ≤ 0. D

Since we have, by condition (15.27), Δ(−ψ) = 1 > 0

in D,

(15.28)

460

15 Path Functions of Markov Processes via Semigroup Theory

it follows from an application of the strong maximum principle (Theorem 10.7) with W := Δ and u := −ψ that ψ(x) ≡ ψ(x0 ) in D. Hence we have the assertion Δ(−ψ) = 0 in D. This contradicts condition (15.28). (2) Next we consider the case: There exists a point x0 ∈ ∂D such that ψ(x0 ) = min ψ ≤ 0. D

Then, by applying the boundary point lemma (Lemma 10.11) with W := Δ to the function u := −ψ we obtain from inequality (15.28) that ∂ψ  (x ) > 0. ∂n 0 This contradicts the boundary condition 0=

∂ψ  ∂ψ  (x ) + 0 (x ) > 0 γ (x0 )ψ(x0 ) ≥ ∂n 0 ∂n 0

on ∂D,

since γ 0(x0 ) ≤ 0 on ∂D. The proof of Claim 15.13 is complete.   Now we choose a real-valued, smooth function θ ∈ C ∞ (R) that satisfies condition (15.22), and define a function f ∈ C ∞ (D) by the formula    |x − x0 | f (x) = 1 − θ ψ(x), ε where 0 < ε < dist (x, ∂D). Then, in view of Claim 15.13 it is easy to verify that f ∈ D(AR ) and satisfies conditions (15.14) of Theorem 15.9. (II) The case where x0 is an arbitrary (boundary) point of ∂D: By change of coordinates, we may assume that  x0 = (0 , 0) = (0, . . . , 0, 0) , (15.23) n = (0 , 1) = (0, . . . , 0, 1) ∈ RN . If we define a function f ∈ C ∞ (D) by the formula   x   |x |  N  θ f (x , xN ) = 1 − θ ψ(x , xN ), ε ε

15.4 Examples of Multi-dimensional Diffusion Processes

461

it follows from conditions (15.22) and (15.27) that ∂f +γ 0(x )f ∂n    x   |x |   ∂ N  = θ 1−θ ψ(x , xN ) ∂xN ε ε xN =0   x   |x |   N  θ +γ 0(x ) 1 − θ ψ  ε ε xN =0          |x | |x | ∂ψ  1   = − θ (0)θ (x , xN ) · ψ(x , 0) + 1 − θ(0)θ ε ε ε ∂xN xN =0     |x | +γ 0(x ) 1 − θ (0) θ ψ(x , 0) ε           |x | |x | ∂ψ  (x , xN ) + 1−θ = 1−θ γ 0(x )ψ(x , 0) ε ∂xN ε xN =0       |x | ∂ψ  = 1−θ +γ 0(x )ψ ε ∂n = 0 on ∂D. This proves that f ∈ D(AR ). Moreover, we have the following three assertions (1), (2) and (3) (see Figures 15.9 and 15.11): (1) Since f (x) = 0 in the ε/2 neighborhood Uε/2 (x0 ) of x0 , it follows from Claim 15.13 that f (y) = AR f (y) = 0 for all y ∈ Uε/2 (x0 ). This proves that condition (15.14c) in Theorem 15.9 is satisfied. (2) It is clear that f (x) ≥ 0 on D, so√that condition (15.14a) is satisfied. (3) Since f (y) = ψ(y) outside the N ε neighborhood U√N ε (x0 ) of x0 , it follows that f (y) > 0 for all y ∈ D \ U√N ε (x0 ). This proves that condition (15.14b) is satisfied. Therefore, by applying Theorem 15.9 to the operator AR we obtain the following theorem: Theorem 15.14 (the Robin case). The strong Markov process {yt (ω)} associated with the Feller semigroup {Tt }t≥0 enjoys the property P ({ω : y t (ω) is continuous for all t ≥ 0}) = 1.

462

15 Path Functions of Markov Processes via Semigroup Theory

15.4.3 The Oblique Derivative Case Finally, we consider the oblique derivative boundary condition BO u =

∂u + β(x ) · u + γ(x )u = 0 on ∂D. ∂n

where n is the unit inward normal vector to the boundary ∂D, γ ∈ C ∞ (∂D), and β(x ) · u :=

γ(x ) ≤ 0 on ∂D, N −1 

βi (x )

i=1

∂u ∂xi

1

is a tangent vector field of class C on the boundary ∂D. We introduce a linear operator AO as follows: (a) The domain D(AO ) is the space   ∂u   D(AO ) = u ∈ C(D) : Δu ∈ C(D), + β(x ) · u + γ(x )u = 0 on ∂D . ∂n (b) AO u = Δu for every u ∈ D(AO ). Then it follows from an application of Theorem 1.6 with L := BO that the operator AO is the infinitesimal generator of a Feller semigroup {Ut }t≥0 with oblique derivative boundary condition. Let {zt (ω)} be the strong Markov process corresponding to the Feller semigroup {Ut }t≥0 . In this subsection we study the path-continuity of the Markov process {zt (ω)}. To do this, we shall make use of Theorem 15.9. If we introduce a vector filed  by the formula  = n + β, then the oblique derivative boundary condition BO u =

∂u + β(x ) · u + γ(x )u = 0 ∂n

on ∂D

can be written in the form (see Figure 15.12 below) ∂u + γ(x )u = 0 ∂

on ∂D.

Furthermore, for each initial point y  = (y1 , y2 , . . . , yN −1 ) ∈ RN −1 we consider the following initial-value problem for ordinary differential equations:

15.4 Examples of Multi-dimensional Diffusion Processes

463

D

n

∂D

β

Fig. 15.12. The oblique derivative boundary condition

⎧ dη  1 ⎪ ⎪ dt = β1 (η(y , t)), ⎪ ⎪ ⎨ .. . dηN −1 ⎪ ⎪ = βN −1 (η(y  , t)), ⎪ ⎪ ⎩ dηdt N dt = 1,

∂u ∂

η1 (y  , 0) = y1 , .. . ηN −1 (y  , 0) = yN −1 , ηN (y  , 0) = 0.

(15.29)

We remark that the initial-value problem (15.29) has a unique local solution η(y  , t) = (η1 (y  , t), η2 (y  , t), . . . , ηN (y  , t)) , since the vector filed β(x ) is Lipschitz continuous. Hence we can introduce a new change of variables by the formula x = (x1 , x2 , . . . , xN ) = (η1 (y  , yN ), η2 (y  , yN ), . . . , ηN (y  , yN )) = η(y) for y = (y  , yN ) ∈ RN . The situation may be represented schematically by Figure 15.13 below. xN

yN

D

RN +

x = η(y)

y

x

Fig. 15.13. The change of variables x = η(y)

Then it follows that the vector field  can be simplified as follows: N −1 N −1   ∂ ∂xj ∂ ∂xN ∂ ∂ ∂ ∂ . = + = βj (x) + = ∂yN ∂yN ∂xj ∂yN ∂xN ∂xj ∂xN ∂ j=1 j=1

Hence we obtain that

464

15 Path Functions of Markov Processes via Semigroup Theory

∂u ∂u + β(x ) · u + γ(x )u = + γ(x )u = ∂n ∂ where





  ∂0 u +γ 0(y  )0 u  , ∂yN yN =0

u 0(y) = u(x), γ 0(y  ) = γ(x ).

In this way, we are reduced to the Robin boundary condition case in the half-space RN +: ∂0 u BR u 0= +γ 0(y  )0 u = 0 on ∂RN +. ∂yN Therefore, by applying Theorem 15.9 to the operator AO we obtain the following theorem: Theorem 15.15 (the oblique derivative case). The strong Markov process {zt (ω)} associated with the Feller semigroup {Ut }t≥0 enjoys the property P ({ω : t (ω) is continuous for all t ≥ 0}) = 1.

15.5 Notes and Comments This chapter is devoted to the study of path functions of Markov processes via the Hille-Yosida semigroup theory. The material is taken from Dynkin [34, Chapter V], Lamperti [74, Chapter 8] and Taira [122, Chapter 12]. Section 15.1: Theorem 15.3 is due to Lamperti [74, Chapter 8, Section 2] and Theorem 15.5 is due to Lamperti [74, Section 8.3, Theorem 1], respectively. Section 15.2: Theorem 15.6 is adapted from Lamperti [74, Section 8.3, Theorem 2] and Corollary 15.7 is adapted from Dynkin [34, Theorem 3.5] (see also [74, Section 8.3, Corollary]). Section 15.3: Theorem 15.9 is taken from Dynkin [34, Theorem 3.9] and Lamperti [74, Section 8.3, Theorem 3]. Section 15.4: Our functional analytic proof of Theorem 15.11 (the Neumann case), Theorem 15.14 (the Robin case) and Theorem 15.15 (the oblique derivative case) may be new.

Part VI

Concluding Remarks

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

This book is devoted to a concise and accessible exposition of the functional analytic approach to the problem of construction of strong Markov processes with Ventcel’ boundary conditions in probability. More precisely, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space D \ M until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (see Figure 16.1 below). Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators which may be considered as a modern version of the classical potential theory.

∂D

D

M = {μ = 0}

Fig. 16.1. A Markovian particle dies when it reaches the set M

More generally, it is known (see [15], [98], [114], [122], [143]) that the infinitesimal generator W of a Feller semigroup {Tt }t≥0 is described analytically by a Waldenfels integro-differential operator W and a Ventcel’ boundary condition L, which we formulate precisely.

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1 16

468

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

16.1 Formulation of the Problem Now let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary. In this chapter we consider a second-order, elliptic integrodifferential operator W with real coefficients such that W u(x) = Au(x) + Su(x) ⎛ ⎞ N N 2   u ∂ ∂u := ⎝ aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ ∂x ∂x ∂xi i j i,j=1 i=1

(16.1)

   N  ∂u i + d(x)u(x) + e (x) (x) + s(x, y) u(y) ∂xi D i=1     N   ∂u α − σα (x, y) u(x) + (χα (y) − χ (x)) (x) dy . j j ∂χα j α=1 j=1

Here: (1) aij (x) ∈ C ∞ (RN ), aij (x) = aji (x) for x ∈ RN and 1 ≤ i, j ≤ N , and there exists a constant a0 > 0 such that N 

aij (x)ξi ξj ≥ a0 |ξ|2

for all (x, ξ) ∈ RN × RN .

i,j=1

bi (x) ∈ C ∞ (RN ) for 1 ≤ i ≤ N . (A1)(x) = c(x) ∈ C ∞ (RN ) and c(x) ≤ 0 and c(x) ≡ 0 in D. d(x) ∈ C ∞ (RN ). ei (x) ∈ C ∞ (RN ) for 1 ≤ i ≤ N . The integral kernel s(x, y) is the distribution kernel of a properly supN ported, pseudo-differential operator S ∈ L2−κ 1,0 (R ) with κ > 0, which has the transmission property with respect to ∂D (see Subsection 5.2), and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in the product space RN × RN . The measure dy is the Lebesgue measure on RN . (7) Let {(Uα , χα )}α=1 be a finite open covering of D by local charts, and let {σα }α=1 be a family of functions in C ∞ (D × D) (see [41, Proposition (8.15)]) such that supp σα ⊂ Uα × Uα ,

(2) (3) (4) (5) (6)

and  

σα (x, y) = 1

α=1

in a neighborhood of the diagonal ΔD = {(x, x) : x ∈ D}.

16.1 Formulation of the Problem

469

(8) The function (S1)(x) ∈ C ∞ (RN ) satisfies the condition  (S1)(x) = d(x) + D

    s(x, y) 1 − σα (x, y) dy ≤ 0

in D.

(16.2)

α=1

The operator W given by formula (16.1) is called a second-order, Waldenfels integro-differential operator (cf. [140], [15]). The differential operator A is called a diffusion operator which describes analytically a strong Markov process with continuous paths (diffusion process) in the interior D. In fact, we remark that the differential operator A is local, that is, the value Au(x0 ) at an interior point x0 ∈ D is determined by the values of u in an arbitrary small neighborhood of x0 . Moreover, it is known from Peetre’s theorem ([88]) that a linear operator is local if and only if it is a differential operator. Therefore, we have an assurance of the following assertion: The infinitesimal generator A of a Feller semigroup {Tt }t≥0 on the state space D is a differential operator in the interior D of D if the paths of its corresponding Markov process are continuous. The operator S is called a second-order, L´evy integro-differential operator which is supposed to correspond to the jump phenomenon in the interior D; a Markovian particle moves by jumps to a random point, chosen with kernel s(x, y), in the interior D. Therefore, the Waldenfels integro-differential operator W is supposed to correspond to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D (see Figure 16.2 below).

D

Fig. 16.2. A Markovian particle moves both by jumps and continuously in D

Intuitively, condition (16.2) implies that the jump phenomenon from a point x ∈ D to the outside of a neighborhood of x in D is “dominated” by the absorption phenomenon at x. We remark that the integral operator   s(x, y) u(y) Sr u(x) = D

470

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

  N  ∂u α α − σα (x, y) u(x) + (χj (y) − χj (x)) α (x) dy ∂χj α=1 j=1  

is a “regularization” of S, since the integrand is absolutely convergent. Indeed, it suffices to note (see Theorem 4.51) that, for any compact K ⊂ RN , there exists a constant CK > 0 such that the distribution kernel s(x, y) of S ∈ N L2−κ 1,0 (R ) with κ > 0, satisfies the estimate |s(x, y)| ≤

CK |x − y|N +2−κ

for all x, y ∈ K and x = y.

We give a simple example of S for κ = 3: N Example 16.1. The symbol p(x, ξ) of S ∈ L−1 1,0 (R ) is given by the formula

  2 1 ξN p(x, ξ) =  exp − 2 , ξ  ξ   where ξ = (ξ  , ξN ),

ξ  = (ξ1 , ξ2 , . . . , ξN −1 ),

ξ   =

; 1 + |ξ  |2 .

It is easy to see that the distribution kernel s(x, y) of S is equal to the following (see Example 4.1, (1)):   1 |x − y  |2 1 |xN − yN |2 s(x, y) = − exp − , 4 |xN − yN |2 2π N/2 |xN − yN |N −1 where x = (x , xN ) = y = (y  , yN ) , x = (x1 , x2 , . . . , xN −1 ) ,

y  = (y1 , y2 , . . . , yN −1 ) .

Let L be a second-order, boundary condition such that, in terms of local coordinates (x1 , x2 , . . . , xN −1 ) on ∂D, Lu(x ) = Γ u(x ) − δ(x ) W u(x ) = Λu(x ) + T u(x ) − δ(x ) W u(x ) (16.3) ∂u  (x ) − δ(x ) W u(x ) + T u(x ) = Qu(x ) + μ(x ) ∂n  N  −1 N −1  ∂2u ij   i  ∂u    := α (x ) (x ) + β (x ) (x ) + γ(x )u(x ) ∂xi ∂xj ∂xi i,j=1 i=1  N −1  ∂u  ∂u  (x ) − δ(x ) W u(x ) + η(x )u(x ) + ζ i (x ) (x ) ∂n ∂xi i=1    r(x , y ) u(y  )

+ μ(x )  + ∂D

16.1 Formulation of the Problem

471

  N −1  ∂u   β β   − τβ (x , y ) u(x ) + (χj (y) − χj (x )) (x ) dy  ∂x j j=1 β=1   + t(x , y) u(y) m 

−





D m 

  N −1    ∂u τβ (x , y) u(x ) + (χβj (y) − χβj (x )) β (x ) dy . ∂χj j=1 β=1

Here: (1) The operator Q is a second-order, degenerate elliptic differential operator on ∂D with non-positive principal symbol. In other words, the αij are  the components of a smooth symmetric contravariant tensor of type 20 on ∂D satisfying the condition N −1 

αij (x )ξi ξj ≥ 0

i,j=1

for all x ∈ ∂D and ξ  = (2) (3) (4) (5) (6)

(7) (8) (9)

(10)

&N −1 j=1

ξj dxj ∈ Tx∗ (∂D).

Here Tx∗ (∂D) is the cotangent space of ∂D at x . (Q1)(x ) = γ(x ) ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D. μ(x ) ∈ C ∞ (∂D) and μ(x ) ≥ 0 on ∂D. δ(x ) ∈ C ∞ (∂D) and δ(x ) ≥ 0 on ∂D. n = (n1 , n2 , . . . , nN ) is the unit inward normal to the boundary ∂D (see Figure 16.1). The integral kernel r(x , y  ) is the distribution kernel of a pseudo-differen  1 tial operator R ∈ L2−κ 1,0 (∂D) with κ1 > 0, and r(x , y ) ≥ 0 off the    diagonal Δ∂D = {(x , x ) : x ∈ ∂D} in the product space ∂D × ∂D. The density dy  is a strictly positive density on ∂D. η(x ) ∈ C ∞ (∂D). &N −1 ζ(x ) = i=1 ζ i (x )∂/(∂xi ) is a smooth tangent vector field on ∂D. The integral kernel t(x, y) is the distribution kernel of a properly supN 2 ported, pseudo-differential operator T ∈ L2−κ 1,0 (R ) with κ2 > 0, which has the transmission property with respect to the boundary ∂D, and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in the product space RN × RN . Let {(Uβ , χβ )}m β=1 be a finite open covering of D by local charts, and m let {τβ }β=1 be a family of functions in C ∞ (D × D) (see [41, Proposition (8.15)]) such that supp τβ ⊂ Uβ × Uβ , and m  β=1

τβ (x, y) = 1

472

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

  in a neighborhood of the diagonal ΔD = (x, x) : x ∈ D . N 2 (11) The fundamental hypothesis concerning T ∈ L2−κ 1,0 (R ) is formulated as follows (see [15, p. 436, Section II.2.3]): The function



t(x , y)

D

m 

N −1   β  2  χj (y) − χβj (x ) dy τβ (x , y) χβN (y) +

is continuous on ∂D. (12) The function (T 1)(x ) ∈ C ∞ (∂D) satisfies the condition    m  (T 1)(x ) = η(x ) + r(x , y  ) 1 − τβ (x , y  ) dy  ∂D

(16.5)

β=1

  m  t(x , y) 1 − τβ (x , y) dy

 + D

≤0

(16.4)

j=1

β=1

β=1

on ∂D.

We remark that the integral operator   Rr u(x ) = r(x , y  ) u(y  ) ∂D m 

−

  N −1  ∂u τβ (x , y  ) u(x ) + (χβj (y) − χβj (x )) β (x ) dy  ∂χj j=1 β=1

is a “regularization” of R, since the integrand is absolutely convergent. Indeed, it suffices to note (see Theorem 4.51) that, for any compact neighborhood Ux  in ∂D there exists a constant CK > 0 such that the distribution kernel r(x , y  ) of R satisfies the estimate |r(x , y  )| ≤

 CK |x − y  |N +1−κ1

for all y  ∈ Ux and y  = x ,

where |x − y  | is the geodesic distance between x and y  with respect to the Riemannian metric of ∂D induced by the natural metric of RN . The operator Rr is called a second-order, Ventcel’–L´evy boundary operator on the boundary. We give a simple example of R for κ1 = 1: √ Example 16.2. Let R = − −Δ ∈ L11,0 (∂D) where Δ is the Laplace–Beltrami operator on the boundary ∂D. Then its principal symbol is equal to −|ξ  | where |ξ  | is the length of ξ  with respect to the Riemannian metric of ∂D. It is easy to see that the distribution kernel r(x , y  ) of R is given by the formula r(x , y  ) =

1 Γ (N/2) π N/2 |x − y  |N

for all x , y  ∈ ∂D and x = y  .

16.1 Formulation of the Problem

473

We give a simple example of T that satisfies condition (16.4) for N = 3 and κ2 = 5: 3 Example 16.3. The symbol q(x, ξ) of T ∈ L−3 1,0 (R ) is given by the formula

q(x, ξ) = where



1 3

ξ  

ξ = (ξ  , ξ3 ),

1−

2ξ32

2

ξ  

ξ  = (ξ1 , ξ2 ),



  ξ2 exp − 3 2 , ξ  

ξ   =

; 1 + |ξ  |2 .

It is easy to see that the distribution kernel t(x, y) of T is equal to the following (see Example 4.1, (1)):   |x − y  |2 |x3 − y3 |2 1 − t(x, y) = 3/2 exp − . 4 |x3 − y3 |2 4π Here

x = (x , x3 ) = y = (y  , y3 ) ,

x = (x1 , x2 ), y  = (y1 , y2 ) .

The boundary condition L given by formula (16.3) is called a second-order, Ventcel’ boundary condition (cf. [143]). The six terms of L N −1 

αij (x )

i,j=1

N −1  ∂2u ∂u  (x ) + β i (x ) (x ), ∂xi ∂xj ∂x i i=1

∂u  (x ), δ(x ) W u(x ), ∂n     m N −1   ∂u  r(x , y  ) u(y  ) − τβ (x, y) u(x ) + (yj − xj ) (x ) dy  , ∂xj ∂D j=1

γ(x )u(x ),

μ(x )

β=1

   m N −1   ∂u     t(x , y) u(y) − τβ (x , y) u(x ) + (yj − xj ) (x ) dy ∂xj D j=1



β=1

are supposed to correspond to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the viscosity phenomenon and the jump phenomenon on the boundary and the inward jump phenomenon from the boundary, respectively (see Figures 16.3 through 16.5 below). Intuitively, condition (16.5) implies that the jump phenomenon from a point x ∈ ∂D to the outside of a neighborhood of x in D is “dominated” by the absorption phenomenon at x . This chapter is devoted to a functional analytic approach to the problem of construction of Feller semigroups with Ventcel’ boundary conditions. More precisely we consider the following problem (see [122, Problem 1.1]): Problem. Conversely, given analytic data (W, L), can we construct a Feller semigroup {Tt }t≥0 whose infinitesimal generator is characterized by (W, L) ?

474

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

D

∂D

D

∂D

absorption

reflection

Fig. 16.3. Absorption and reflection phenomena

D

∂D

D

∂D

diffusion along the boundary

viscosity

Fig. 16.4. Diffusion along ∂D and viscosity phenomenon

D

∂D

D

∂D

jump into the interior

jump on the boundary

Fig. 16.5. Jump phenomena into D and on ∂D

Table 16.1 below gives a bird’s-eye view of Markov processes, Feller semigroups and elliptic boundary value problems and how these relate to each other: We shall only restrict ourselves to some aspects that have been discussed in the work [114] through [122]. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. It focuses on the relationship between two interrelated subjects in analysis; Feller semigroups and elliptic boundary value problems, providing powerful methods for future research.

16.2 Statement of Main Results

Probability (Microscope)

Functional Analysis (Macroscope)

Elliptic Boundary Value Problems (Mesoscope)

Markov process X = (xt )

Feller semigroup {Tt }t≥0

Infinitesimal generators W0 , W

Markov transition function pt (·, dy)

Tt f =

 D

pt (·, dy) f (y)

475

Tt = e t W 0 or Tt = et W

Chapman and Kolmogorov equation

Semigroup property Tt+s = Tt · Ts

Waldenfels operator W=A+S

Various diffusion phenomena

Function  spaces  C  D  C0 D \ M

Ventcel’ (Wentzell) condition L

Table 16.1. A bird’s-eye view of Markov processes, Feller semigroups and boundary value problems (Theorems 16.1 through 16.4)

16.2 Statement of Main Results 16.2.1 The Transversal Case First, we generalize Theorem 14.1 to the transversal case. We say that the boundary condition L is transversal on the boundary ∂D if it satisfies the condition  t(x , y) dy = +∞ if μ(x ) = δ(x ) = 0. (16.6) D

The intuitive meaning of condition (16.6) is that a Markovian particle jumps away “instantaneously” from the points x ∈ ∂D where neither reflection nor viscosity phenomenon occurs (which is similar to the reflection phenomenon). Probabilistically, this means that every Markov process on the boundary ∂D is the “trace” on ∂D of trajectories of some Markov process on the closure D = D ∪ ∂D (see Figure 16.6 below). The next theorem asserts that there exists a Feller semigroup on the state space D corresponding to such a diffusion phenomenon that one of the re-

476

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

∂D

D

Fig. 16.6. The probabilistic meaning of transversality condition

flection phenomenon, the viscosity phenomenon and the inward jump phenomenon from the boundary occurs at each point of the boundary ∂D (see [122, Theorem 1.2]): Theorem 16.1. We define a linear operator W0 : C(D) −→ C(D) in the space C(D) as follows: (a) The domain of definition D (W0 ) is the space   D (W0 ) = u ∈ C(D) : W u ∈ C(D), Lu = 0 on ∂D .

(16.7)

(b) W0 u = W u for every u ∈ D (W0 ). Here W u and Lu are taken in the sense of distributions. Assume that the boundary condition L is transversal on the boundary ∂D. Then the operator W0 generates a Feller semigroup  t W0  e t≥0 on the state space D. A probabilistic meaning of Theorem 16.1 may be represented schematically by Figure 16.7 below. We remark that Theorem 16.1 was proved by Taira [114, Theorem 10.1.3] under some additional conditions, and also by Cancelier [24, Th´eor`eme 3.2] and [116, Theorem 1]. On the other hand, Takanobu and Watanabe [130] proved a probabilistic version of Theorem 16.1 in the case where the domain D is the half space RN + (see [130, Corollary]). Our functional analytic approach to strong Markov processes in the transversal case may be visualized as in Figure 16.8 below.

16.2 Statement of Main Results

477

∂D

D

Fig. 16.7. A probabilistic meaning of Theorem 16.1 (transversal case)

{Tt } : Feller semigroup on C(D)

pt (x, ·) : uniform stochastic continuity

+

right-continuous Markov process

(Xt ) : strong Markov process

Feller property

Fig. 16.8. The functional analytic approach to strong Markov processes in the transversal case

16.2.2 The Non-Transversal Case Secondly, we generalize Theorem 14.8 to the non-transversal case. To do so, we assume that: (G.1) There exists a second-order, Ventcel’ boundary condition Lν such that Lu = m(x ) Lν u + γ(x ) u

on ∂D.

(16.8)

(G.2) m(x ) − γ(x ) > 0 on ∂D. Here: (2) γ(x ) ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D. (3 ) m(x ) ∈ C ∞ (∂D) and m(x ) ≥ 0 on ∂D, and Lν is given, in terms of local coordinates (x1 , x2 , . . . , xN −1 ), by the formula Lν u(x ) = Γ u(x ) − δ(x ) W u(x ) = Λ u(x ) + T u(x ) − δ(x ) W u(x ) ∂u  (x ) + T u(x ) − δ(x ) W u(x ) = Q u(x ) + μ(x ) ∂n

478

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

N −1 

 N −1  ∂2u i  ∂u   := α (x ) (x ) + β (x ) (x ) ∂xi ∂xj ∂xi i,j=1 i=1  ∂u + μ(x ) (x ) − δ(x ) W u(x ) + η(x )u(x ) ∂n   N −1  i ∂u  ζ (x ) (x ) + r(x , y  ) u(y  ) + ∂xi ∂D i=1    m N −1   ∂u     τβ (x , y ) u(x ) + (x ) dy  − yj − xj ∂x j j=1 β=1   + t(x , y) u(y) D

−

m  β=1

ij



    N −1  ∂u   τβ (x , y ) u(x ) + (x ) dy , yj − xj ∂xj j=1 



and it satisfies the condition Γ 1(x ) = T 1(x ) = η(x ) +

  m  r(x , y  ) 1 − τβ (x , y  ) dy 

 ∂D

β=1

  m  t(x , y) 1 − τβ (x , y) dy

 + D

≤0

β=1

on ∂D.

Moreover, we assume the transversality condition for Lν  t(x , y) dy = +∞ if μ(x ) = δ(x ) = 0. D

We let

(16.5 )

       M = x ∈ ∂D : μ(x ) = δ(x ) = 0, t(x , y) dy < ∞ . D

Then, by condition (16.6 ) it follows that M = {x ∈ ∂D : m(x ) = 0} , since we have the formulas μ(x ) = m(x ) μ(x ), Q = m(x ) Q,

δ(x ) = m(x ) δ(x ),

t(x , y) = m(x ) t(x , y).

(16.6 )

16.2 Statement of Main Results

479

Therefore, we find that the boundary condition L, defined by formula (16.8), is not transversal on ∂D. The intuitive meaning of conditions (G.1) and (G.2) is that a Markovian particle does not stay on ∂D for any period of time until it “dies” at the time when it reaches the set M where the particle is definitely absorbed. Now we introduce a subspace of C(D) which is associated with the boundary condition L. By condition (G.2), we find that the boundary condition Lu = m(x ) Lν u + γ(x ) u = 0

on ∂D

includes the condition u = 0 on M . With this fact in mind, we let     C0 D \ M = u ∈ C(D) : u = 0 on M .   The space C0 D \ M is a closed subspace of C(D); hence it is a Banach space.   A strongly continuous semigroup {Tt }t≥0 on the space C0 D \ M is called a Feller semigroup on thestate space  D\M if it is non-negative and contractive on the Banach space C0 D \ M :   f ∈ C0 D \ M and 0 ≤ f ≤ 1 on D \ M =⇒ 0 ≤ Tt f ≤ 1 on D \ M . We define a linear operator     W : C0 D \ M −→ C0 D \ M   in the space C0 D \ M as follows: (a) The domain of definition D (W) is the space D (W) (16.9)       = u ∈ C0 D \ M : W u ∈ C0 D \ M , Lu = 0 on ∂D . (b) Wu = W u for every u ∈ D (W). The next theorem is a generalization of Theorem 16.1 to the non-transversal case (see [122, Theorem 1.3]): Theorem 16.2. Assume that conditions (G.1) and (G.2) are satisfied. Then the operator W defined by formula (16.9) generates a Feller semigroup  t W e t≥0 on the state space D \ M .

480

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

Theorem 16.2 was proved by Taira [114, Theorem 2] under some additional conditions. We remark that Taira [116] has proved Theorem 16.2 in the case where Lν = ∂/∂n and δ(x ) ≡ 0 on ∂D, by using the Lp theory of pseudodifferential operators (see [116, Theorem 4]). If Tt is a Feller semigroup on the state space D \ M , then there exists a unique Markov transition function pt (·, dy) on D \ M such that  pt (x, dy)f (y) for all f ∈ C0 (D \ M ), Tt f (x) = D\M

and that pt (·, dy) is the transition function of some strong Markov process. On the other hand, the intuitive meaning of conditions (G.1) and (G.2) is that the absorption phenomenon occurs at each point of the set M = {x ∈ ∂D : m(x ) = 0} . Therefore, Theorem 16.2 asserts that there exists a Feller semigroup on the state space D\M corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D \ M until it “dies” at which time it reaches the set M . The situation of Theorem 16.2 may be represented schematically by Figure 16.9 below.

∂D

D

M = {m = 0} Fig. 16.9. A probabilistic meaning of Theorem 16.2 (the non-transversal case)

Our functional analytic approach to strong Markov processes in the nontransversal case may be visualized as in Figure 16.10 below. 16.2.3 The Lower Order Case Finally, we consider the case where all the operators S, T and R are pseudodifferential operators of order less than one. Then we can take σ(x, y) ≡ 1 on D × D, and write the operator W in the form (16.1 ):

16.2 Statement of Main Results

481

{Tt } : Feller semigroup on C0 (D \ M )

pt (x, ·) : uniform stochastic continuity

right-continuous Markov process

+

C0 -property

X : strong Markov process

Fig. 16.10. The functional analytic approach to strong Markov processes in the non-transversal case

W u(x) = Au(x) + Su(x) (16.1 )   N N  ∂2u ∂u ij i a (x) (x) + b (x) (x) + c(x)u(x) := ∂xi ∂xj ∂xi i,j=1 i=1    + d(x)u(x) + s(x, y)[u(y) − u(x)]dy . D

Here: (3) (A1)(x) = c(x) ∈ C ∞ (RN ) and c(x) ≤ 0 and c(x) ≡ 0 in D. (6 ) The integral kernel s(x, y) is the distribution kernel of a properly supN ported, pseudo-differential operator S ∈ L1−κ 1,0 (R ) with κ > 0, which has the transmission property with respect to the boundary ∂D, and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in the product space RN × RN .  ) The function (S1)(x) = d(x) ∈ C ∞ (RN ) satisfies the condition (8 (S1)(x) = d(x) ≤ 0 in D. Similarly, the boundary condition L can be written in the form (16.3 ): Lu(x ) = Γ u(x ) − δ(x ) W u(x ) = Λu(x ) + T u(x ) − δ(x ) W u(x ) (16.3 ) ∂u = Qu(x ) + μ(x ) (x ) − δ(x ) W u(x ) + T u(x ) ∂n  N  −1 N −1  ∂2u ij   i  ∂u    := α (x ) (x ) + β (x ) (x ) + γ(x )u(x ) ∂xi ∂xj ∂xi i,j=1 i=1 ∂u + μ(x ) (x ) − δ(x ) W u(x ) ∂n  + η(x )u(x )

482

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

 +

r(x , y  )[u(y  ) − u(x )]dy  +

∂D

 t(x , y)[u(y) − u(x )]dy .

 D

Here: (Q1)(x ) = γ(x ) ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D. μ(x ) ∈ C ∞ (∂D) and μ(x ) ≥ 0 on ∂D. δ(x ) ∈ C ∞ (∂D) and δ(x ) ≥ 0 on ∂D. The integral kernel r(x , y  ) is the distribution kernel of a pseudo  1 differential operator R ∈ L1−κ 1,0 (∂D) with κ1 > 0, and r(x , y ) ≥ 0    off the diagonal {(x , x ) : x ∈ ∂D} in the product space ∂D × ∂D. (9 ) The integral kernel t(x, y) is the distribution kernel of a properly supN 2 ported, pseudo-differential operator T ∈ L1−κ 1,0 (R ) with κ2 > 0, which has the transmission property with respect to the boundary ∂D, and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in the product space RN × RN .  (12 ) The function (T 1)(x ) = η(x ) ∈ C ∞ (∂D) satisfies the condition (2) (3) (4) (6 )

(T 1)(x ) = η(x ) ≤ 0 on ∂D. Then Theorems 16.1 and 16.2 may be simplified as follows (see [122, Theorems 1.4 and 1.5]): Theorem 16.3. Assume that the operator W and the boundary condition L are of the forms (16.1 ) and (16.3 ), respectively. If the boundary condition L is transversal on the boundary ∂D, then the operator W0 defined by formula (16.7) generates a Feller semigroup {et W0 }t≥0 on the state space D. Theorem 16.4. Assume that the operator W and the boundary condition L are of the forms (16.1 ) and (16.3 ), respectively. If conditions (G.1) and (G.2) are satisfied, then the operator W defined by formula (16.9) generates a Feller semigroup {et W }t≥0 on the state space D \ M . Theorems 16.1, 16.2, 16.3 and 16.4 solve from the viewpoint of functional analysis the problem of construction of Feller semigroups with Ventcel’ boundary conditions for elliptic, Waldenfels integro-differential operators. Finally, we give an overview of general results on generation theorems for Feller semigroups proved mainly by the author using the theory of pseudodifferential operators in Table 16.2 below ([57], [105], [106]) and the Calder´ on– Zygmund theory of singular integral operators ([23], [108]): Here (see Definitions 11.1 and 11.5): W = A + S,

(16.1) 



L = Γ − δ(x ) W = (Λ + T ) − δ(x ) W    ∂ = μ(x ) + Q + T − δ(x ) W. ∂n

(16.3)

16.2 Statement of Main Results

diffusion operator A

L´evy operator S

elliptic smooth case

S≡0

Ventcel’ condition L = Γ − δ(x ) W ∂ Λ = μ(x ) ∂n +Q

T ≡0 ∂ Λ = μ(x ) ∂n +Q Q = γ(x ) (Robin case) T ≡0

using the theory of

proved by

pseudodifferential operators

[114]

pseudodifferential operators

[117]

elliptic smooth case

convex domain

elliptic smooth case

general case

general case

pseudodifferential operators

[116] [122]

elliptic discontinuous case

general case

μ(x ) ≡ 0 Q = γ(x ) ≡ 1 (Dirichlet case) T ≡0

singular integral operators

[118] [127]

αij ≡ 0 (oblique derivative) (case) T ≡0

singular integral operators

[120] [129]

elliptic discontinuous case

S≡0

483

Table 16.2. An overview of generation theorems for Feller semigroups

In [118], [120] and [127], we prove existence theorems for Feller semigroups with Dirichlet boundary condition, oblique derivative boundary condition and first-order Ventcel’ boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. Our approach there is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calder´ on–Zygmund theory of singular integral operators with non-smooth kernels ([28], [29], [79]). It should be emphasized that the Calder´ on–Zygmund theory of singular integral operators with non-smooth kernels provides a powerful tool to deal with smoothness of solutions of elliptic boundary value problems, with minimal as-

484

16 The State-of-the-Art of Generation Theorems for Feller Semigroups

sumptions of regularity on the coefficients. The theory of singular integrals continues to be one of the most influential works in modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited (see [23], [109]).

16.3 Notes and Comments We state a brief history of the stochastic analysis methods for Ventcel’ boundary value problems. More precisely, we remark that Ventcel’ boundary value problems are studied by Anderson [9], [10], Cattiaux [25] and Takanobu– Watanabe [130] from the viewpoint of stochastic analysis (see also Ikeda– Watanabe [62, Chapter IV, Section 7]). (I) Anderson [9] and [10] studies the non-degenerate case under low regularity in the framework of the submartingale problem and shows the existence and uniqueness of solutions to the considered submartingale problem. (II) Takanobu–Watanabe [130] study certain cases of both degenerate interior and boundary operators under minimal assumptions of regularity based on the theory of stochastic differential equations, and they show the existence and uniqueness of solutions. Such existence and uniqueness results on the diffusion processes corresponding to the boundary value problems imply the existence and uniqueness of the associated Feller semigroups on the space of continuous functions. (III) Cattiaux [25] studies the hypoellipticity for diffusions with Ventcel’ boundary conditions. By making use of a variant of the Malliavin calculus under H¨omander’s type conditions ([58]), he proves that some laws and conditional laws of such diffusions have a smooth density with respect to the Lebesgue measure.

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Index

Ap -completely continuous, 413, 418 A-diffusion, 375, 377 a priori estimate, 11, 27, 240, 265, 267, 269, 273, 274, 418 absorbing barrier Brownian motion, 71, 88 absorption phenomenon, 8, 361 adjoint, 148, 152, 288, 290 adjoint operator, 288, 290, 291 Agmon’s method, 27, 273, 275, 281 algebra of pseudo-differential operators, 148 amplitude, 140, 141, 143, 147 analytic semigroup, 12, 14, 35, 39, 292, 399, 437 analytic semigroup via the Cauchy integral, 35 Ascoli–Arzel` a theorem, 118, 411 associated semigroup, 83 asymptotic expansion, 137 Avogadro’s number, 2

Banach space, 106, 243 Banach’s open mapping theorem, 404 barrier, 311, 317 basic properties of Fourier integral operators, 141 basic properties of pseudo-differential operators, 142 Besov space, 9, 114–116, 233, 241 Besov space boundedness theorem, 150, 406

Besov space boundedness theorem of pseudo-differential operators on a manifold, 152 Bessel potential, 112, 115, 143, 155, 170, 173, 188 Bessel potential space, 113 bijective, 284, 288, 378, 391, 404 Bolzano–Weierstrass theorem, 118 Borel kernel, 297, 299, 300, 304 Borel measurable, 67, 68, 298 Borel measure, 54 Borel set, 65, 66, 68, 439 boundary, 297 boundary condition, 8, 235, 242, 375 boundary maximum principle, 325 boundary operator, 202, 325 boundary point lemma, 316, 372, 379, 413, 454, 460 boundary value problem, 235 boundary value problem via the Boutet de Monvel calculus, 242 boundary value problem with spectral parameter, 266 bounded continuous function, 77 bounded operator, 160, 362, 363, 370, 379, 392 bounded perturbation, 94 Boutet de Monvel algebra, 23, 192 Boutet de Monvel calculus, 244, 245, 377, 390 Brownian motion, 70, 87 Brownian motion with absorbing barrier, 71, 88

© Springer Nature Switzerland AG 2020 K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, https://doi.org/10.1007/978-3-030-48788-1

494

Index

Brownian motion with constant drift, 70, 87 Brownian motion with reflecting barrier, 71, 87 Brownian motion with sticking barrier, 72, 87 Brownian motion with sticky barrier, 87 C0 -function, 78 C0 -property, 78 C0 transition function, 78, 79, 83 Calder´ on–Zygmund operator, 143 Cauchy density, 71 Cauchy process, 71, 87 Cauchy’s theorem, 37, 41, 43, 44, 47 cemetery, 68 Chapman–Kolmogorov equation, 68 characterization of the resolvent set of the Dirichlet problem, 257 classical elliptic pseudo-differential operator, 231 classical potential, 209 classical pseudo-differential operator, 268, 277, 283, 284, 290 classical pseudo-differential operator, 26, 232, 237 classical pseudo-differential operator on a manifold, 152 closable, 89, 431 closed extension, 89, 96, 251, 253, 367, 372, 428, 432, 433, 436 closed graph theorem, 285 closed linear operator, 238, 239, 345 closed operator, 238, 239, 274, 282–284, 290, 345 closed range theorem, 288, 292 codimension, 160, 282, 283 coercive, 10 commutator, 353 compact, 85 compact manifold, 104, 123, 230 compact manifold with boundary, 468 compact metric space, 62, 63, 94, 442, 449, 450 compact operator, 118, 256, 285–289 compact perturbation, 29, 256, 403, 407, 410 compactification, 346

compactly supported, 51 complete symbol, 147–149 completed π-topology tensor product, 187, 191 completely continuous operator, 118, 256 complex parameter, 266, 275 composition of potential and trace operators, 202 composition of pseudo-differential operators, 148 composition of pseudo-differential operators on a manifold, 152 conditional expectation, 65 conditional probability, 65 conormal derivative, 8 conservative, 68, 457 continuous function, 50 continuous injection, 243 continuous operator, 160 continuous path, 73, 74, 446, 450 contractive, 14, 83 converge weakly, 63 convolution, 204 convolution kernel, 171 cotangent bundle, 150, 230, 266 cotangent vector, 153 definition of a classical pseudodifferential operator, 148 definition of a classical symbol, 137 definition of a Fourier integral distribution, 140 definition of a Fourier integral operator, 141 definition of a phase function, 138 definition of a pseudo-differential operator, 142, 143 definition of a pseudo-differential operator on a manifold, 151 definition of a symbol, 136 definition of an elliptic pseudodifferential operator, 149 definition of an oscillatory integral, 140 definition of stopping times, 74 definition of strong Markov property, 76 degenerate boundary condition, 16 degenerate boundary value problem, 9 degenerate problem, 10

Index dense, 364 dense range, 89, 374, 428 densely defined, 35, 84, 89, 238, 239, 282–284, 290, 345, 382, 428, 435 densely defined operator, 84, 89, 238, 239, 345, 428, 435 density, 104, 232 diagonal, 144, 145, 151, 299, 468 diameter, 411 diffusion coefficient, 6, 361 diffusion operator, 6, 361, 469 diffusion process, 70, 79 Dini’s theorem, 384, 385 Dirac measure, 110, 170 Dirichlet condition, 8, 195, 199, 388 Dirichlet form, 255 Dirichlet problem, 176, 194, 230, 233, 249, 267, 276, 361, 362 Dirichlet-to-Neumann operator, 26, 202, 208, 237, 246, 248, 249, 251, 253, 268, 276, 283, 284 distribution, 13, 18, 368, 453 distribution kernel, 142, 147, 188, 252, 305, 468, 470–473, 481, 482 distribution kernel of a pseudodifferential operator, 154 domain (of definition), 12, 13, 18, 19, 345, 346, 367, 369–371, 375, 377, 378, 389, 418, 428 dominated convergence theorem, 44, 107 double, 123, 230, 266 double layer potential, 210 double of a domain, 23, 104, 116, 179, 190, 213, 230 double of a manifold, 123 drift, 70, 87 drift coefficient, 6, 361 dual space, 109, 113, 115 Dynkin’s theorem, 83 elliptic boundary value problem, 207, 266 elliptic differential operator, 27, 266, 273, 275 elliptic pseudo-differential operator, 232

495

elliptic pseudo-differential operator on a manifold, 153 elliptic symbol, 137 elliptic Waldenfels integro-differential operator, 5, 307, 310, 316, 401, 482 elliptic Waldenfels integro-differential operator and maximum principle, 297 elliptic Waldenfels operator, 5, 307, 310, 316, 401, 482 elliptic Waldenfels operator and maximum principle, 297 equivalent (Markov process), 442 event, 64 existence and uniqueness theorem, 404 existence and uniqueness theorem for the Dirichlet problem, 233, 361 existence and uniqueness theorem for the Neumann problem, 233 existence theorem for Feller semigroups, 476, 479, 482 existence theorem for the Dirichlet problem, 233, 361 existence theorem for the Neumann problem, 233 exit time, 180 expectation, 65 Feller function, 77 Feller (transition) function, 449 Feller property, 77 Feller semigroup, 14, 79, 83, 96, 251, 253, 449, 473, 476, 479, 482 Feller semigroup for Waldenfels integro-differential operators, 345 Feller semigroup for Waldenfels operators, 345 Feller semigroup with reflecting barrier, 376 first exit time, 16, 403 formulation of the boundary value problem, 234 formulation of the problem, 473 Fourier integral operator, 136, 141 Fourier inversion formula, 109, 205 Fourier transform, 107, 110 Fr´echet space, 106 Fredholm boundary operator, 246

496

Index

Fredholm (closed) operator, 282, 284 Fredholm integral equation, 236, 405 Fredholm operator, 160 Friedrichs mollifier, 132, 364, 369 Fubini’s theorem, 37, 212, 440, 443 function rapidly decreasing at infinity, 108 function space, 104 functional calculus for the Laplacian via the heat kernel, 170 Gagliardo–Nirenberg inequality, 117, 348 general Besov space, 116 general boundary operator, 28 general existence theorem for Feller semigroups, 360 general Robin boundary operator, 207, 242 general Robin problem, 207 general Sobolev space, 116 generation theorem for analytic semigroups, 10, 12, 13, 18, 19, 39 generation theorem for Feller semigroups, 14, 20, 84, 88, 428, 437 geodesic distance, 252, 306, 472 global regularity theorem for the Dirichlet problem, 361 graph norm, 418 Green operator, 25, 84, 85, 234, 246, 362, 377, 389 Green representation formula, 212 harmonic operator, 249, 253, 362, 378 Hausdorff–Young inequality, 227 heat kernel, 22, 108, 155, 170 Hille–Yosida theorem, 84, 436 Hille–Yosida theory, 86 Hille–Yosida–Ray theorem, 28, 88, 297, 428 H¨ older continuous, 105, 189 H¨ older inequality, 415, 416 H¨ older norm, 403, 407, 409 H¨ older space, 16, 29, 105, 106, 189, 190, 249, 310, 361, 370, 402, 404, 406 homogeneous principal symbol, 148, 149, 153 homotopy, 256

homotopy invariant, 256 Hopf boundary point lemma, 316, 372, 379, 413 Hopf’s boundary point lemma, 454, 460 H¨ ormander class, 143, 154, 406 hypoelliptic, 154, 289, 291 hypoelliptic pseudo-differential operator, 154 index formula of Agranoviˇc–Dynin type, 244, 246 index of a Fredholm (closed) operator, 283 index of a Fredholm operator, 160 infinitesimal generator, 84, 85, 89, 94, 96, 156, 251, 253, 361, 374, 377, 389, 433, 450, 473 infinitesimal generator of a Feller semigroup, 450 initial distribution of a stochastic process, 440 initial value problem, 170 injective, 282 injectivity, 381, 394 inner regular, 57 integro-differential operator, 302, 304, 306, 307 interior, 297 interior normal, 267 interior regularity theorem for the Dirichlet problem, 361 invariance of pseudo-differential operators under change of coordinates, 149 inverse Fourier transform, 107, 110 inverse local time, 97, 250 inward normal, 8, 234, 361, 453 isomorphism, 248, 404 Jordan decomposition, 53 jump density, 6 jump discontinuity, 442 jump formula, 135, 212, 213 jump phenomenon, 6 kernel, 142 Lp boundedness of pseudo-differential operators, 150 Lp space, 106

Index Lp theory of pseudo-differential operators, 103 Laguerre polynomial, 183 λ-dependent localization, 28, 30, 345, 349, 404, 419, 421 Laplace transform, 86, 170 Laplacian, 249, 268, 277 large jump, 7, 308 Lebesgue dominated convergence theorem, 107, 336 Lebesgue’s dominated convergence theorem, 44 L´evy integro-differential operator, 6, 410, 469 L´evy integro-differential operator, 403 L´evy kernel, 334, 335, 337 L´evy measure, 155, 156 lifetime, 68 local operator, 469 local time, 97, 250, 251 local unity function, 299, 334, 336–339 localized Besov space, 116 localized Sobolev space, 116 locally compact metric space, 50, 73, 76, 78, 439, 442 locally H¨ older continuous, 105 Lopatinskii–Shapiro condition, 9 lower order case, 480 manifold, 151, 266, 290 manifold with boundary, 5, 104, 123, 213, 230, 266, 325, 360 Markov process, 15, 49, 64, 67, 68, 439, 441, 442 Markov property, 3, 66, 69, 440 Markov time, 74 Markov transition function, 15, 68, 440, 441 maximum norm, 12, 407, 430, 449 maximum principle, 88, 297, 412, 428 maximum principle for elliptic Waldenfels integro-differential operators, 307 maximum principle for elliptic Waldenfels operators, 307 measurable, 65, 67, 68 measurable stochastic process, 440, 442, 446 measure, 50

497

metric space, 50, 65, 66, 68, 85, 89 minimal closed extension, 20, 89, 96, 251, 253, 367, 372, 428, 432, 433, 436 modified Bessel function, 113, 188 mollifier, 132, 364, 369 moment condition, 6, 307, 402 negative variation measure, 53 Neumann boundary condition, 452 Neumann boundary value problem, 252 Neumann case, 457 Neumann condition, 8, 196, 200, 202, 388, 389 Neumann problem, 233, 234, 405 Neumann series, 92, 94 Newtonian potential, 110, 143, 155, 170, 172, 209, 231 non-degenerate, 10 non-homogeneous Neumann problem, 245 non-homogeneous Robin problem, 242, 244 non-negative, 14, 83, 85, 362, 363, 370, 379, 392 non-negative linear functional, 54 non-negative measure, 299 non-positive principal symbol, 303 non-transversal boundary condition, 477, 479, 482 non-transversal case, 477, 479 norm, 9, 106, 107, 113–116 normal coordinate, 104 normal Markov transition function, 442 normal transition function, 69, 73 null space, 160, 233, 267, 276, 282, 283, 285, 291 oblique derivative boundary condition, 462 oblique derivative case, 464 one-point compactification, 51, 76, 346 operator norm, 63 oscillatory integral, 139, 155 outward normal, 25, 234 parameter, 266 parametrix, 27, 149, 154, 265, 269, 406 Parseval formula, 109

498

Index

partition of unity, 120, 299 path, 65, 440 path continuity, 73 path-continuity, 442, 449 path function, 73 Peetre’s inequality, 226 Peetre’s lemma, 284 phase function, 138, 140, 141, 143 Plancherel theorem, 111 point at infinity, 51, 76, 346 Poisson integral formula, 210 Poisson kernel, 170, 176, 194, 232, 233 Poisson operator, 26, 193, 199, 200, 202, 233, 249, 253, 267, 276, 378 Poisson process, 70, 86 positive Borel kernel, 299 positive boundary maximum principle, 328 positive boundary maximum principle for the Ventcel’–L´evy boundary operator, 328 positive boundary maximum principle the Ventcel’ boundary operator, 329 positive maximum principle, 28, 297, 303, 334 positive parameter, 362 positive variation measure, 53 positively homogeneous, 136 potential operator, 191–193, 245 potential symbol, 192 principal part, 137 principal symbol, 148–150, 153, 245, 251 probabilistic convolution semigroup, 156 probability measure, 64 probability space, 64 progressively measurable, 76 projective topology tensor product, 187 Prokhorov’s theorem, 99 properly supported, 145 pseudo-differential operator, 103, 141, 304, 482 pseudo-differential operator on a manifold, 151 pseudo-local property, 145, 180 quotient topology, 346

Radon measure, 6, 54, 307, 402 random variable, 64, 65, 439 rapidly decreasing, 108 rational function, 214 real measure, 52 reflecting barrier Brownian motion, 71, 87 reflecting diffusion, 95, 97, 249, 377 reflection phenomenon, 8, 361, 377 regularity property, 239 regularity theorem for the Dirichlet problem, 361 regularizer, 145, 152 Rellich–Kondrachov theorem, 287, 289 Rellich–Kondrachov theorem, 118, 256, 418 residue theorem, 37, 43, 174, 194, 211 resolvent, 12, 14, 35, 85, 282, 356 resolvent equation, 363, 364, 371 resolvent set, 12, 14, 35, 255, 282, 356 restriction, 140, 180, 182 Riemannian metric, 268 Riesz kernel, 110 Riesz operator, 143 Riesz potential, 110, 142, 155, 170, 173 Riesz representation theorem, 55 Riesz–Markov representation theorem, 50, 53, 62, 63 right continuous, 73, 74 right continuous path, 73, 74 right-continuous Markov process, 78, 442, 445, 449 right-continuous σ-algebras, 74 right-inverse, 247 Robin boundary condition, 458 Robin boundary operator, 244 Robin case, 461 σ-algebra, 66 σ-algebra of all Borel sets, 76, 439 σ-algebra of Borel sets, 65, 66, 68 sample point, 64 sample space, 65, 440 Schwartz kernel theorem, 142 Schwartz space, 108, 113, 115 second-order, elliptic integro-differential operator, 468 sectional trace, 124

Index sectional trace theorem, 125 Seeley extension operator, 119 Seeley extension theorem, 118 Seeley’s extension operator, 229 semigroup, 36, 49 semigroup property, 36 seminorm, 106, 108, 115, 116 seminorm of a symbol, 136 separable, 64, 73, 76, 78, 439 signed measure, 52 single and double layer potentials, 209 single layer potential, 209 singular Green operator, 180, 191, 198 singular Green symbol, 198 singular integral operator, 482 singular support, 140 Slobodecki˘ı seminorm, 115 small jump, 7, 308 Sobolev space, 9, 112, 116, 249 Sobolev space of fractional order, 113 Sobolev’s imbedding theorem, 117, 345 space of bounded continuous functions, 77 space of bounded linear operators, 36 space of continuous functions, 50, 367, 449 space of signed measures, 52 special reduction to the boundary, 235 spectral analysis of the Dirichlet eigenvalue problem, 254 spectral parameter, 27, 254, 273, 362 stable process, 155 state space, 65, 440 sticking barrier Brownian motion, 72, 87 sticky barrier Brownian motion, 87 stochastic process, 65, 439 stochastic process governed by the transition function, 440–442 stopping time, 74, 76 strong Markov process, 15, 76, 78, 449 strong Markov property, 76, 78 strong maximum principle, 310, 323, 454, 460 strongly continuous, 79, 80, 83 strongly continuous semigroup, 79, 83 sub-σ-algebra, 440 support, 51, 300 support condition, 6, 402

499

supremum norm, 77, 407, 422, 430 surface and volume potentials, 213 surface potential, 214, 232 surjective, 282 surjectivity, 281 symbol, 136 symbol class, 136 symmetric α-stable process, 155, 156 symmetric contravariant tensor, 330, 471 symmetric stable process, 155 tempered distribution, 109 terminal point, 68, 72, 77, 88 termination coefficient, 6, 361 Tietze’s extension theorem, 50 time change, 251 topological complement, 285, 286 total variation, 53 total variation measure, 53 trace, 234, 243 trace map, 122, 124, 207, 235 trace operator, 191, 194, 195, 245, 267, 276 trace symbol, 194 trace theorem, 122, 124, 234, 243 trajectory, 65, 440 transition function, 3, 15, 67 transition map, 150 transmission property, 179, 180, 185–191 transpose, 138, 139, 148, 149, 152 transversal boundary condition, 475, 478, 482 transversal case, 475 trap, 73 uniform motion, 70, 86 uniform operator topology, 36 uniform stochastic continuity, 78 uniformly elliptic differential operator, 230, 242, 360 uniformly stochastically continuous, 78, 79, 83, 449 unique solvability of the Dirichlet problem, 255 uniqueness theorem for the Dirichlet problem, 233, 361

500

Index

uniqueness theorem for the Neumann problem, 233 Urysohn’s lemma, 50 v.p. (valeur principal), 112, 156 vanish at infinity, 51 Ventcel’ boundary operator, 330 Ventcel’ boundary condition, 7, 361, 473 Ventcel’ boundary operator, 325, 329 Ventcel’ boundary value problem, 367 Ventcel’ kernel, 326, 327, 330, 340 Ventcel’–L´evy boundary operator, 28, 325–328, 330, 340, 472 Ventcel’–Viˇsik boundary operator, 330 volume potential, 228, 231

Waldenfels integro-differential operator, 5, 6, 307, 310, 316, 401, 467, 469, 482 Waldenfels operator, 5, 6, 307, 310, 316, 401, 467, 469, 482 weak convergence of measures, 63 weak maximum principle, 308, 362, 372, 385, 413 weak topology of measures, 78 weak* convergence, 64 Wiener measure, 3 Wiener–Hopf technique, 181, 203 Yosida approximation, 85 zero extension, 16, 23, 180, 182, 190, 195, 213, 403

LECTURE NOTES IN MATHEMATICS

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Editors in Chief: J.-M. Morel, B. Teissier; Editorial Policy 1. Lecture Notes aim to report new developments in all areas of mathematics and their applications – quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. Manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialised lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this “lecture notes” character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series, though habilitation theses may be appropriate. 2. Besides monographs, multi-author manuscripts resulting from SUMMER SCHOOLS or similar INTENSIVE COURSES are welcome, provided their objective was held to present an active mathematical topic to an audience at the beginning or intermediate graduate level (a list of participants should be provided). The resulting manuscript should not be just a collection of course notes, but should require advance planning and coordination among the main lecturers. The subject matter should dictate the structure of the book. This structure should be motivated and explained in a scientific introduction, and the notation, references, index and formulation of results should be, if possible, unified by the editors. Each contribution should have an abstract and an introduction referring to the other contributions. In other words, more preparatory work must go into a multi-authored volume than simply assembling a disparate collection of papers, communicated at the event. 3. Manuscripts should be submitted either online at www.editorialmanager.com/lnm to Springer’s mathematics editorial in Heidelberg, or electronically to one of the series editors. Authors should be aware that incomplete or insufficiently close-to-final manuscripts almost always result in longer refereeing times and nevertheless unclear referees’ recommendations, making further refereeing of a final draft necessary. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Parallel submission of a manuscript to another publisher while under consideration for LNM is not acceptable and can lead to rejection. 4. In general, monographs will be sent out to at least 2 external referees for evaluation. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. Volume Editors of multi-author works are expected to arrange for the refereeing, to the usual scientific standards, of the individual contributions. If the resulting reports can be

forwarded to the LNM Editorial Board, this is very helpful. If no reports are forwarded or if other questions remain unclear in respect of homogeneity etc, the series editors may wish to consult external referees for an overall evaluation of the volume. 5. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include – a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader. – For evaluation purposes, manuscripts should be submitted as pdf files. 6. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files (see LaTeX templates online: https://www.springer.com/gb/authors-editors/book-authorseditors/manuscriptpreparation/ 5636) plus the corresponding pdf- or zipped ps-file. The LaTeX source files are essential for producing the full-text online version of the book, see http://link.springer.com/bookseries/304 for the existing online volumes of LNM). The technical production of a Lecture Notes volume takes approximately 12 weeks. Additional instructions, if necessary, are available on request from [email protected]. 7. Authors receive a total of 30 free copies of their volume and free access to their book on SpringerLink, but no royalties. They are entitled to a discount of 33.3 % on the price of Springer books purchased for their personal use, if ordering directly from Springer. 8. Commitment to publish is made by a Publishing Agreement; contributing authors of multiauthor books are requested to sign a Consent to Publish form. Springer-Verlag registers the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: ´ Professor Jean-Michel Morel, CMLA, Ecole Normale Sup´erieure de Cachan, France E-mail: [email protected] Professor Bernard Teissier, Equipe G´eom´etrie et Dynamique, Institut de Math´ematiques de Jussieu – Paris Rive Gauche, Paris, France E-mail: [email protected] Springer: Ute McCrory, Mathematics, Heidelberg, Germany, E-mail: [email protected]